Institutionen för systemteknik Department of Electrical Engineering Examensarbete Steering System Modelling for Heavy Duty Vehicles Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet av Rickard Sjölund and Nicklas Vedin LiTH-ISY-EX--15/4852--SE Linköping 2015 Department of Electrical Engineering Linköpings universitet SE-581 83 Linköping, Sweden Linköpings tekniska högskola Linköpings universitet 581 83 Linköping Steering System Modelling for Heavy Duty Vehicles Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet av Rickard Sjölund and Nicklas Vedin LiTH-ISY-EX--15/4852--SE Handledare: Gustav Lindmark isy, Linköping University Jonny Andersson Scania Examinator: Martin Enqvist isy, Linköping University Linköping, 26 June 2015 Avdelning, Institution Division, Department Datum Date Avdelningen för Reglerteknik Department of Electrical Engineering SE-581 83 Linköping 2015-06-26 Språk Language Rapporttyp Report category ISBN Svenska/Swedish Licentiatavhandling ISRN Engelska/English Examensarbete C-uppsats D-uppsats — LiTH-ISY-EX--15/4852--SE Serietitel och serienummer Title of series, numbering Övrig rapport ISSN — URL för elektronisk version http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-119770 Titel Title Modellering av styrsystem för tunga fordon Författare Author Rickard Sjölund and Nicklas Vedin Steering System Modelling for Heavy Duty Vehicles Sammanfattning Abstract Framtidens tunga fordon kommer att designas och tillverkas med förbättrade förarstödsfunktioner, adas. Vid utveckling av adas förenklas utvecklingsprocessen av en modell över fordonets dynamik. Ett system med stor påverkan på fordonets laterala dynamik och utvecklingen av adas är styrsystemet. Syftet med detta examensarbete är att utveckla en tillförlitlig och simuleringseffektiv modell av ett tungt fordons styrsystem. Insignalen i systemet är ett moment på ratten och den resulterande hjulvinkeln betraktas som utsignal. Med hjälp av bindningsgrafer och kända samband utvecklas fysikaliska modeller av systemets komponenter. Vissa komponenter modelleras med olinjära effektivitetsförluster och friktion av olika komplexitet. Verktyg som linjär regression och olinjär sökning i rutnät används för att skatta okända parametrar och funktioner utifrån mätdata. De olika delsystemen identifieras separat i den utsträckning det anses möjligt. Olika modelleringsalternativ övervägs, valideras, och jämförs. Fördelar och nackdelar med olika modelleringsval diskuteras. Slutligen väljs en olinjär tillståndsmodell för dess höga precision och beräkningseffektivitet. Eftersom den slutgiltiga modellen kan användas för att simulera ett tungt fordons styrsystem på en skrivbordsdator snabbare än realtid uppfyller den sitt syfte. Nyckelord Keywords Advanced Driver Assistance System, Steering System, Lane Keep Assist, Power Steering Gear, Physical Modelling, Hydraulic Power Assist, Bond Graph Sammanfattning Framtidens tunga fordon kommer att designas och tillverkas med förbättrade förarstödsfunktioner, adas. Vid utveckling av adas förenklas utvecklingsprocessen av en modell över fordonets dynamik. Ett system med stor påverkan på fordonets laterala dynamik och utvecklingen av adas är styrsystemet. Syftet med detta examensarbete är att utveckla en tillförlitlig och simuleringseffektiv modell av ett tungt fordons styrsystem. Insignalen i systemet är ett moment på ratten och den resulterande hjulvinkeln betraktas som utsignal. Med hjälp av bindningsgrafer och kända samband utvecklas fysikaliska modeller av systemets komponenter. Vissa komponenter modelleras med olinjära effektivitetsförluster och friktion av olika komplexitet. Verktyg som linjär regression och olinjär sökning i rutnät används för att skatta okända parametrar och funktioner utifrån mätdata. De olika delsystemen identifieras separat i den utsträckning det anses möjligt. Olika modelleringsalternativ övervägs, valideras, och jämförs. Fördelar och nackdelar med olika modelleringsval diskuteras. Slutligen väljs en olinjär tillståndsmodell för dess höga precision och beräkningseffektivitet. Eftersom den slutgiltiga modellen kan användas för att simulera ett tungt fordons styrsystem på en skrivbordsdator snabbare än realtid uppfyller den sitt syfte. iii Abstract Future heavy duty vehicles will be designed and manufactured with improved Advanced Driver Assistance Systems, adas. When developing adas, an accurate model of the vehicle dynamics greatly simplifies the development process. One element integral to the vehicle lateral dynamics and development of adas is the steering system. This thesis aims to develop an accurate model of a heavy duty vehicle steering system suitable for simulations. The input to the system is an input torque at the steering wheel and the output is the wheel angle. Physical models of the system components are developed using bond graphs and known relations. Some components are modelled with non-linear inefficiencies and friction of different complexity. Unknown parameters and functions are identified from measurement data using system identification tools such as, for example, linear regression and non-linear grid search. The different subsystems are identified separately to the extent deemed possible. Different model designs are considered, validated, and compared. The advantages and disadvantages of different model choices are discussed. Finally, a nonlinear state space model is selected for its high accuracy and efficiency. As this final model can be used to simulate a heavy duty vehicle steering system on a desktop computer faster than real time, it fulfills its purpose. v Acknowledgments This work has been performed at the department Driver Assistance Controls at Scania, where we want to thank Jonny and Daniel for their interest, support, and commitment. Thanks to Christoffer, Joseph, Linus, and Markus for your guidance towards a solution. Our work would not have been possible without the support of Emil, Jolle, and Krystof, thank you for your help providing us with measurement data. Additionally we want to thank all other staff involved in our work, as well as our department at Scania for an enjoyable experience. Many thanks to Gustav for your hard work interpreting unclear drafts of the report and contributing with input on how to improve the quality of our work. We would also like to thank Martin for his inputs throughout the process, not least the insights he provided early in the planning phase leading to a more structured and specific goal. Finally, we want to thank our friends and family for your support throughout our university studies and an unforgettable experience. Södertälje, June 2015 Rickard Sjölund and Nicklas Vedin vii Contents Notation xiii 1 Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . 1.2 Problem Formulation . . . . . . . . . . . . . . . . . 1.3 Aim of the Thesis . . . . . . . . . . . . . . . . . . . 1.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Delimitations . . . . . . . . . . . . . . . . . . . . . . 1.6 Previous Work . . . . . . . . . . . . . . . . . . . . . 1.6.1 Previous Models of Mechanics . . . . . . . . 1.6.2 Previous Models of Hydraulics . . . . . . . 1.6.3 Previous Models of Friction and Hysteresis 1.6.4 Previous Models of External forces . . . . . 1.6.5 Practical Basis for the Modelling . . . . . . 1.7 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 2 3 3 3 4 4 5 5 5 2 System Description 2.1 Mechanics . . . . . . . . . . 2.1.1 Steering Column . . 2.1.2 Power Steering Gear 2.1.3 Steering Linkage . . 2.2 Hydraulics . . . . . . . . . . 2.3 External Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 7 8 9 10 10 3 General Modelling and Simulation Techniques 3.1 Model Types . . . . . . . . . . . . . . . . . . 3.2 Physical Modelling . . . . . . . . . . . . . . 3.2.1 Bond Graphs . . . . . . . . . . . . . . 3.3 System Identification . . . . . . . . . . . . . 3.4 Parameter Estimation . . . . . . . . . . . . . 3.5 Simulation and Solvers . . . . . . . . . . . . 3.5.1 Discretization of State Space Models 3.5.2 Differential Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 14 14 14 16 16 17 17 18 . . . . . . . . . . . . ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Contents 3.5.3 Solvers in Simulink . . . . . . . . . . . . . . . . . . . . . . . 4 Steering System Mechanics 4.1 Fundamental Mechanical Principles . . . . . . . . . . . . 4.1.1 Moment of Inertia Model . . . . . . . . . . . . . . . 4.2 Steering Column . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Previously Developed Steering Column Models . . 4.2.2 Proposed Steering Column Model . . . . . . . . . . 4.3 Power Steering Gear . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Previously Developed Power Steering Gear Models 4.3.2 Proposed Power Steering Gear Model . . . . . . . . 4.4 Steering Linkage . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Previously Developed Steering Linkage Models . . 4.4.2 Proposed Steering Linkage Model . . . . . . . . . . 4.5 Combined State Space Model . . . . . . . . . . . . . . . . 4.6 Wheel Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . 21 21 22 22 23 23 25 25 25 29 29 30 31 33 5 Steering System Hydraulics 5.1 Hydraulic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Valve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 41 6 Data Analysis and Parameter Estimation 6.1 Measurement Data . . . . . . . . . . . . . . . . . . . 6.1.1 Data Set 1 – Mechanical Efficiency . . . . . . 6.1.2 Data Set 2 – Valve Characteristics . . . . . . . 6.1.3 Data Set 3 – Hydraulic Efficiency . . . . . . . 6.1.4 Data Set 4 – Input Analysis . . . . . . . . . . 6.1.5 Data Set 5 – Reversible Torque . . . . . . . . . 6.1.6 Data Set 6 – Stationary Vehicle Measurements 6.2 Parameters and Functions . . . . . . . . . . . . . . . 6.2.1 Steering Column . . . . . . . . . . . . . . . . 6.3 Valve Characteristics . . . . . . . . . . . . . . . . . . 6.3.1 Valve Offset . . . . . . . . . . . . . . . . . . . 6.3.2 Valve Function . . . . . . . . . . . . . . . . . . 6.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Efficiency Constants . . . . . . . . . . . . . . 6.4.2 Adaptation of Hydraulic Valve Coefficients . 6.4.3 Hammerstein-Wiener Models . . . . . . . . . 6.5 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Linear Friction Model . . . . . . . . . . . . . . 6.5.2 Non-linear Friction Models . . . . . . . . . . 6.5.3 Proposed Friction Model . . . . . . . . . . . . 6.6 External Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 44 44 46 46 46 46 49 49 49 51 51 55 57 60 61 62 62 64 66 67 7 Results 7.1 Static Behaviour and Hysteresis . . . . . . . . . . . . . . . . . . . . 7.1.1 Static Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 70 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Contents 7.1.2 Efficiency Constants . . . . . . . . . . . . . . . . . . . 7.1.3 Hydraulic Adaptation . . . . . . . . . . . . . . . . . . 7.1.4 Wiener Model . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Comparison of Hysteresis Models for Static Data . . . 7.2 Dynamic Behaviour and Full Vehicle Steering System Model 7.2.1 Comparison of Hysteresis Models for Dynamic Data . 7.2.2 Comparison of Models for the Jacking Torque . . . . . 7.2.3 Comparison of Friction Models . . . . . . . . . . . . . 7.2.4 Comparison of Solvers . . . . . . . . . . . . . . . . . . 7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Summary 8.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Suggestions for Model Accuracy Improvement 8.2.2 Model Expansion . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 73 73 76 76 77 77 81 85 85 . . . . . . . . . . . . 89 89 90 90 91 93 Notation Acronyms Acronym adas lka ode dae hpas mse nrmse dof Meaning Advanced Driver Assistance Systems Lane Keep Assist Ordinary differential equation Differential Algebraic Equation Hydraulic Power Assisted Steering Mean Square Error Normalized Root Mean Square Error Degrees Of Freedom xiii xiv Notation Nomenclature Mechanics Notation bsc eof f Fdl Fdl,j Fg,p Ftf ,n Ftf ,r Fz fhyd fl,p fµ,p g hµ,g hµ,kp hµ,sc hµ,wh hsus Jda Meaning Unit Steering column damping coefficient Lateral offset at the hub Drag link force Draglink force induced by jacking torque Gravitational force acting on the piston Force acting on the piston from the worm Force acting on the piston from the sector shaft Front axle normal force Hydraulic force Piston loss force Piston friction Gravitational acceleration constant Wheel to ground friction King pin friction Steering column friction Wheel friction Suspension height Drop arm moment of inertia [N ms/rad] [m] [N ] [N ] [N ] [N ] [N ] Jlsc Lower steering column moment of inertia Jss Sector shaft moment of inertia Jsw Steering wheel moment of inertia Jusc Upper steering column moment of inertia Jwh Wheel moment of inertia Jwo kdl ksc ktb lda ldla mdl meq Worm moment of inertia Drag link spring stiffness Steering column spring stiffness Torsion bar spring stiffness Drop arm length Drag link arm length Drag link mass Equivalent mass of the sector shaft, piston and worm Piston mass Track rod mass Worm to piston gear ratio Sector shaft radius Total aligning torque Drop arm torque Steering column string stiffness torque Torque acting on the wheels from the drag link mp mtr n r Ta Tda Tk,sc Tin,wh [N ] [N ] [N ] [N h ] i m/s2 [N m] [N m] [N m] [N m] [m] h i kgm2 /rad h i kgm2 /rad h i kgm2 /rad h i kgm2 /rad h i kgm2 /rad h i kgm2 /rad h i kgm2 /rad [N m/rad] [N m/rad] [N m/rad] [m] [m] [kg] [kg] [kg] [kg] h i m−1 [m] [N m] [N m] [N m] [N m] xv Notation Notation Tis Tj TJwh Tsa Tsc Ttf ,n Ttf ,r Tsw Ttb Tdla vp xp δ δwh,l θK P I θpsg τl,is ωda ωlsc ωss ωusc ωwh ωwo Meaning Unit Input shaft torque Jacking torque Torque acting on the moment of inertia of the wheels Self aligning torque Steering column torque Torque acting on the worm from the piston Torque acting on the sector shaft from the piston Steering wheel torque Torsion bar torque Drag link arm torque Piston velocity Piston position Vehicle steering angle Left wheel angle Kingpin inclination angle Mounted angle of the power steering gear Input shaft loss torque Drop arm angular velocity Lower steering column angular velocity Sector shaft angular velocity Upper steering column angular velocity Wheel angular velocity Worm angular velocity [N m] [N m] [N m] [N m] [N m] [N m] [N m] [N m] [N m] [N m] [m/s] [m] [rad] [rad] [rad] [rad] [N m] [rad/s] [rad/s] [rad/s] [rad/s] [rad/s] [rad/s] xvi Notation Nomenclature Hydraulics Notation Ap Meaning Exposed piston area Ap,L Left side exposed piston area Amin Minimal valve area opening Ap,R Right side exposed piston area Av Valve area A0 cq k1 k3 Valve area opening with 0 input Flow coefficient Valve area parameter Valve area parameter pl Load pressure pp Pump pressure pp,min Minimum pump pressure pp,max Maximum pump pressure pL Left side load pressure pR Right side load pressure p0 Pump return pressure Ttb0 Qf Torsion bar torque that delivers the minimum pump pressure Flow of fluid Ql Qp Unit h i m2 h i m2 h i m2 h i m2 h i m2 h i m2 [−] h[1/N m] i 1/(N m)3 i h N /m2 i h N /m2 i h N /m2 h i N /m2 h i N /m2 h i N /m2 i h N /m2 h i m3 /s h m3 /s i Flow of fluid over the piston h m3 /s i Pump flow h m3 /s i VL Left side chamber volume h m3 /s i VR Right side chamber volume h m3 /s i V0 φ ψ ρ Chamber volume at valve equilibrium Valve angle Pressure drop over an orifice Oil density h i m3 /s [rad] [P h a] i kg/m3 1 Introduction 1.1 Background Vehicle manufacturers and researchers across the world are developing autonomous driving to improve traffic safety and increase fuel efficiency. Autonomous driving requires Advanced Driver Assistance Systems (adas) that integrate both lateral and longitudinal control of the vehicle. To develop adas with increased functionality and reliability, good information is needed about the vehicle dynamics and how the vehicle is affected by different operations such as steering, applied throttle, or brakes. Scania is a manufacturer of heavy vehicles such as buses and trucks. For Scania to be competitive in this market it is becoming increasingly important with driver assistance systems, leading to low fuel consumption and traffic safety. Considering the importance of adas, contributing to this development is an interesting challenge. An example of an adas is Lane Keep Assist, lka. This is a system that helps the driver stay in the intended lane. Vehicles can be equipped with sensors that allow control units to detect where the lane is headed. By using this information and a controller, the system can adjust the torque being applied to the steering system to help the driver follow the lane and thereby increase comfort and avoid collisions. 1.2 Problem Formulation A good model of the vehicle’s lateral motion is a prerequisite for multiple adas and fully autonomous vehicles. The problem treated in this thesis is designing an 1 2 1 Introduction accurate vehicle model for the lateral dynamics of heavy duty vehicles. The focus is on the physical steering system from an input torque on the steering wheel to the output angle of the wheels. 1.3 Aim of the Thesis As noted previously, development of driver assistance functionality requires extensive knowledge of vehicle dynamics; lka specifically requires insight into the steering and lateral dynamics. This project aims to investigate the steering system of a heavy duty vehicle and to design mathematical models representing the system. To maximize the usage of the knowledge, the modelling is performed using varying methods and complexity. The resulting models are evaluated for accuracy and computability. The final objective of the modelling process is developing an accurate model that can be simulated efficiently on a desktop computer. 1.4 Method As the first part of the work process, previous work in the area is studied in order to provide useful information and methods, as well as to learn about the type of models previously developed. Both mathematical models as well as models developed in the programs Simulink and Simscape are studied. The different components that make up the steering system are modelled and validated individually before they are merged to an integrated model. By first developing individual subsystem models of different complexity, the importance of each subsystem is assessed. A visual representation of the modelling methodology can be seen in Figure 1.1. Throughout the development process, physical models are studied to gain a good understanding of the equations that constitute the system. After establishing a theoretical basis for the modelling, the system is divided into appropriate subsystems that can be modelled individually with suitable estimation and validation data. The model development is iterative for each subsystem; the subsystem model is validated and evaluated before selecting one or several acceptable models that can be used in an integrated model. Finally, the result of the model validation is used to compare the models. Figure 1.1: The flow of the method used when designing models. 1.5 Delimitations 1.5 3 Delimitations Due to the size and complexity of the system some delimitations are necessary. All components cannot be modelled in full detail and complex components are replaced with simplified representations – e.g. the dependence of the hydraulic pump on engine speed is omitted. Considering that the main objective of the thesis is modelling the steering system, from the steering wheel to the wheels, little effort is put into modelling the load from the truck and the road. To make use of measurement data from the total truck system either a new simple simulation model or an existing simulation model is used. These simulation models for the external dynamics provide the steering model with a suitable input and use the modelled steering system to simulate the vehicle’s motion. As steering systems vary between truck models, one tractor configuration is chosen for this modelling assignment. The model design is performed with a generalizable and parameterizable implementation to as large extent as possible. It is furthermore assumed that the lateral and the longitudinal motions are decoupled. 1.6 Previous Work Obviously this is not the first model of a steering system developed, and this thesis builds on previous work in both academic and corporate contexts. This section briefly presents previous models that are referenced later in the text or have provided inspiration. As this thesis aims to model the steering system of a heavy duty vehicle, previous work in this area is the most relevant. Both Tagesson [2014] and Dell’Amico [2013] give extensive descriptions of steering system for heavy duty vehicles. Models for the same system are also developed in Mårtensson [2005], Dell’Amico and Krus [2015], and Tai et al. [2004]. Many steering system models have also been developed for light automotive vehicles. Even though light vehicle steering systems differ from heavy vehicle ditto, the main functionality is the same. Consequently, much can be learned regarding, for example, the steering column and friction by studying literature regarding light vehicle steering systems. Light vehicle steering system studies used for this purpose are Badawy et al. [1999], Chul Shin et al. [2013], Rösth [2007], Ancha et al. [2007], Qiu et al. [2001], Pfeffer et al. [2008], Wu et al. [2015], and Viana et al. [2010] 1.6.1 Previous Models of Mechanics In Dell’Amico [2013] a review of different approaches to model the steering system mechanics is given. As stated in the review, the previously developed models have between 2 and 5 degrees of freedom – i.e. between 2 and 5 moments of inertia or masses. For example Badawy et al. [1999] uses a complex 4 dof model 4 1 Introduction while Tai et al. [2004], Dell’Amico and Krus [2015] and Chul Shin et al. [2013] prefer simpler two moment of inertia models. Rösth [2007] develops one 2 dof model for the steering system of a light automotive vehicle but chooses to add a third dof to a model for the steering system of a heavy duty vehicle due to the latter’s greater flexibility in the steering linkage. 1.6.2 Previous Models of Hydraulics In addition to the mechanical connection of the steering system, heavy duty vehicles generally provide Hydraulic Power Assisted Steering, hpas, to reduce the steering torque required from the driver. Manufacturers of hpas strive to construct power assist systems with a good trade-off between assistance and road feel for the driver. Furthermore, the fuel consumption is affected by the hydraulics, especially since the pump flow is commonly designed to be close to constant. To further complicate the modelling of power assist systems, the valve characteristics are difficult to model and are kept secret by the manufacturers. The active safety features as well as energy efficiency of the hpas and how these can be improved are discussed in Rösth [2007]. The power steering is investigated using both linear and non-linear modelling techniques. In order to model these systems, the valve design and corresponding characteristics are explained. A slightly more complex model is given in Dell’Amico and Krus [2015] and Dell’Amico [2013] which considers the compressibility of the hydraulic fluid. The compressibility of the hydraulic fluid is also taken into account in Nilsson [2014] where an electrohydraulic system at Scania is studied and modelled. Even though Nilsson [2014] models an open circuit hydraulic system, the methods can be used to model closed circuit electrohydraulic actuator systems such as the hydraulic assist. A model of a hpas is also modelled using bond graphs in Viana et al. [2010]. Furthermore, a model for the hpas system of Scania trucks and suggestions for future improvements is given in Mårtensson [2005]. 1.6.3 Previous Models of Friction and Hysteresis Another important phenomenon to model in the steering system is the friction. In the 2 dof model developed in Dell’Amico and Krus [2015], viscous friction is added to both the upstream mass and downstream mass. Dell’Amico and Krus [2015] also adds static friction to the upstream mass and pressure dependent coulomb friction to the downstream mass. However, Dell’Amico and Krus [2015] later neglects the coulomb friction when linearizing the model. In Qiu et al. [2001] the downstream friction is modelled as a combination of viscous and coulomb friction while Wu et al. [2015] approximates all friction in the steering system with one constant coulomb friction. Like Dell’Amico and Krus [2015], Tagesson [2014] models viscous friction, however Tagesson [2014] argues that the simple coulomb friction should be replaced by a more complex friction model developed in Pfeffer et al. [2008]. Just as the pressure dependent coulomb friction suggested by Dell’Amico and Krus [2015], this model is developed to take the hysteresis into account. 1.7 Outline of the Report 1.6.4 5 Previous Models of External forces In addition to the driver input torque, the steering system is affected by external forces from the tire-road interaction via the wheels. This self aligning torque is decribed in Section 2.3 but models for these torques are developed in: Bastow et al. [2004], Hsu [2009], Tagesson [2014], Wong [2008], and Wu et al. [2015]. 1.6.5 Practical Basis for the Modelling The models proposed in this thesis are influenced in parts by models previously used at Scania. Primarily two models have constituted the foundation and source of inspiration; the ADAMS model and the RT model. At Scania there is a model covering the entire truck; the steering part of this model is henceforth referred to as the ADAMS model. The ADAMS model is a physical model in three dimensions implemented using the multibody modelling language ADAMS. This model is presented in Ikjema [1999]. Another simulation model that is currently used at Scania is the RT model. This model is implemented in Simulink and includes dynamics for the entire truck. However, the steering system in this model is represented by a static relationship including a look-up table. Although the RT model only models the steering system schematically it provides some inspiration when modelling the load and wheel dynamics. 1.7 Outline of the Report This introductory chapter is followed by Chapter 2, where the reader is given a basic understanding of the system that is modelled and how the components interact. The reader is then introduced to the modelling theory in Chapter 3 that is used throughout the thesis. A special focus is given to the distinction between system identification and physical modelling. After providing a general introduction, Chapter 4 contains modelling of mechanical components. In this chapter the mechanical connection in the steering system is modelled based on physical principles. To further explain the behavior of the power steering gear, Chapter 5 contains modelling of the hydraulic system. The hydraulic power assist in the steering system is modelled based on physical principles. Before presenting the system identification methods used in Chapter 6, the measurement data analyzed in the thesis is presented. In the chapter, the measurement data is used to identify unknown parameters and relations in the physical models from Chapter 4 and Chapter 5. Furthermore, model structures are suggested for unknown functions and the measurement data is used when estimating the parameters in the model structures. Previously developed models are compared and validated in Chapter 7. Models are compared with respect to accuracy and computability. The final section of 6 1 Introduction the thesis, Chapter 8, contains discussions concerning the results and proposals of future work. 2 System Description This section provides a description of the steering system that is being modelled. The components that make up the system transfer the input torque on the steering wheel to the wheels and provide power assist. An overview of the system can be seen in Figure 2.1. 2.1 Mechanics In this section the mechanical connection between steering wheel and wheels is discussed. The mechanical connection consists of the steering column, the mechanical connection of the power steering gear and the steering linkage. 2.1.1 Steering Column The steering wheel is attached to the top of the steering column. The mechanism of the steering column allows a mechanical torque to be transferred from the steering wheel to the power steering gear. Upper Steering Column The upper steering column is directly attached to the steering wheel and is also attached to the cab. The lower end is attached to a joint that allows the applied torque to be transferred to the lower steering column. The upper steering column has a certain degree of flexibility. Lower Steering Column Torque is transferred to the power steering gear through the lower steering column and hence the output torque of the lower steering column is the input torque to the power steering gear. As the upper steering column is attached to the cab 7 8 2 System Description Upper steering column Lower steering column Power steering gear Drop arm Drag link Drag link arm Track rod Figure 2.1: Steering system being modelled. Source: Scania and the lower steering column is attached to the chassis some translational flexibility is needed; the cab and the chassis are not rigidly connected. This is accomplished by allowing the lower steering column to be elongated or compressed. 2.1.2 Power Steering Gear In the power steering gear the torque is transferred from the steering column to the steering linkage. As the name implies, this system also amplifies the torque using a hydraulic system powered by the engine. Another important function of the power steering gear is the gear ratio which is approximately 1:20; the gear ratio depends non-linearly on the driver torque and steering angle. An overview of the power steering gear is presented in Figure 2.2. Following the steering column, the applied torque is transferred to the input shaft of the power steering gear. The input shaft operates on a torsion bar that transfers the torque to a worm; the torsion bar basically acts as a spring. The worm is connected via a recirculating ball mechanism to a piston. The piston in turn is connected to the sector shaft and drop arm of the steering linkage. In addition to transferring the torque, the torsion bar acts as a rotor valve that controls the support from the hydraulic servo. The hydraulic force is exerted on 2.1 9 Mechanics the piston and is described more in Section 2.2. It is important that the steering mechanism works even if the hydraulics fail. To fulfill this criteria, the steering box can also be controlled mechanically without additional support from the hydraulics. Oil feed Valve Oil return Worm Torsion bar Input shaft Piston Sector shaft Figure 2.2: Power steering gear. Source: Robert Bosch Automotive Steering GmbH 2.1.3 Steering Linkage The steering linkage transfers the output torque from the power steering gear to the steering knuckle rotating around the king pin of the wheels. The sector shaft of the power steering gear is directly connected to the drop arm, which means that these bodies can be modelled as a single body with a joint moment of inertia. The torque that acts on these bodies is then transferred to a translational force on the drag link. Since the drag link is attached at the kingpin to the wheel through the drag link arm which is attached to the steering knuckle, these bodies can be modelled as a single body. The purpose of the drag link arm and steering knuckle is to convert the translational force of the drag link to a rotational motion of the stub axle that is connected to the wheel. The steering knuckle is also connected to the track rod and the suspension. The track rod connects the two wheels by using ball joints 10 2 System Description that are attached to their respective steering knuckles. As a result, the left and right wheels have similar steering angles; there is an offset which increases with the steering angle and is related to Ackermann Steering Geometry but this will not be modelled in this thesis. 2.2 Hydraulics In this section the hydraulic system in the power steering gear is described. The system is depicted in Figure 2.2. The main purpose of the hydraulics is to apply an additional force to the piston of the power steering gear to ease the lateral handling of the vehicle. To achieve this, the power steering gear is required to sense when a torque is applied. When a torque is applied, a lateral motion is conducted. This can be sensed and controlled with a flow device, a valve located between the torsion bar and the input shaft. The hydraulics of the power steering gear consist of an oil reservoir with a filter, a pump to move the oil at an approximately constant flow rate, as well as an oil flow regulator. The studied hydraulic system is described in ZF Lenksysteme [2015] and its components are modelled accordingly. When a torque is applied at the input shaft of the power steering gear, the result is an angular displacement of the valve. An enlarged image of the valve can be seen in the upper left corner of Figure 2.2. This leads to the connection of ports in the valve housing to channel high or low pressured oil. Inside the housing of the steering box, this pressure difference that comes from the pump and is controlled by the valve and the oil flow regulator affects the piston that moves translationally. The pressure creates an additional force on the piston, in addition to the mechanical force applied from the steering column, by the simple relation that a pressure that acts on an area creates a force. There is a chamber on each side of the piston to allow assisted pressure in both directions. When the torque on the steering wheel ceases, the twisted torsion bar makes the control groves of the valve return to their neutral position. The neutral position of the control groves leads to an equal pressure being applied in both chambers on the side of the piston. The high and low pressures in the hydraulic system are the results of a pump powered by the engine. To limit the influence of varying motor engine speeds the hydraulic system uses the oil flow regulator to acquire a suitable flow rate. The hydraulics are also built with a pressure release valve to prevent excessively high pressures applied on the piston and a hydraulic steering limiter. 2.3 External Behavior The lateral motion of a vehicle is influenced by several external parameters such as wind, road bank, and self-aligning torque. In this thesis these influences are not modelled in detail. However, in order to understand how these external 2.3 External Behavior 11 forces affect the expected steering output, some understanding is helpful. In this section a rudimentary explanation is given for how the self-aligning torque affects the vehicle. Whenever there is a difference between the direction of travel of the vehicle and the wheel plane, denominated a slip angle, a self-aligning torque acts on the wheels and the steering system. This torque originates from the tire-road interaction and acts to decrease the slip angle by rotating the wheel plane towards the direction of travel. The torque depends on a number of parameters such as the weight of the load, the lateral stiffness of the tire, the tire contact length and the velocity of the vehicle. For example, assuming that a vehicle is on a track that has a constant turning radius as well as a constant and high velocity. The vehicle is then affected by a side force. The tires of the vehicle need to produce an opposing cornering force to stay on the track. This cornering force results in a torque acting to rotate the wheel plane towards the direction of travel. Furthermore, this self-aligning torque is what makes the steering wheel return to its original position when driving straight at high velocity. There is also a self-aligning torque resulting from the vertical displacement of the truck at non-zero steering angles. This torque is a function of the caster angle and king pin inclination. The caster angle is the forward rotation of the king pin, the camber angle is the angle between a vertical plane and the wheel plane, and the king pin inclination is the angle of the king pin relative to the vertical plane. These three angles are shown in Figure 2.3. To distinguish between the self-aligning torque caused by slip angles and selfaligning torques caused by the vertical displacements, the former is denominated total aligning torque and the latter is denominated jacking torque. 12 2 Caster angle System Description King pin inclination Camber angle Vertical plane Vertical plane Vertical plane Figure 2.3: Caster angle, camber angle and king pin inclination. The figure is inspired by Bastow et al. [2004]. 3 General Modelling and Simulation Techniques The input to the system described in Chapter 2, is a torque applied on the steering wheel. The output is the angle of the wheels that results in a change of heading. There are disturbances that affect the system output. Examples of such disturbances are wind, track geometry and component deviations. In Ljung [1999] models are described as assumed relationships between different variables. However, this thesis is mainly concerned with mathematical models where relations between variables are stated using mathematical expressions. As stated in Ljung [1999] models can fundamentally be developed in two different ways: either by physical modelling or by system identification. Physical modelling implies splitting the complete system into subsystems and using previous experience to establish relationships between the variables. Previous experience may consist of physical laws and physical principles among other established relationships. The model of the complete system is obtained by connecting the subsystem models using mathematical relations. Models developed by modelling using laws of nature are called white box models. System identification is another approach to building a model. In this case the model is obtained by analyzing system inputs and outputs without considering their physical significance. Using system identification, models can be developed of systems that are not well-understood; these are called black box models. Physical modelling and system identification are combined in grey box modelling. This is accomplished by first modelling using the laws of nature and subsequently using system identification to identify parameters and model dynamics that are not well-understood. The resulting models are based on physical underpinnings but some parameters and dynamics have no physical interpreta13 14 3 General Modelling and Simulation Techniques tion. 3.1 Model Types Mathematical models can be structured in different ways with different advantages and disadvantages. In this section state space models and more generally differential algebraic equation models are described. A differential algebraic equation (dae) is, as defined by Ljung and Glad [2004], any equation on the general form F(ż, z, u) = 0 y = h(z, u), (3.1) where F is a vector valued function. Please note that the equation y = h(z, u) could have been included in F but is written separately for notational convenience. An important special case of dae:s is the state space model given by ẋ = f (x, u) . y = h(x, u). (3.2) and is explained in Ljung and Glad [2004]. This model structure is well studied and having equations on this form can greatly facilitate analysis and simulation. For a general dae the index is given by the number of times the dae has to be differentiated to arrive at a state space model. 3.2 Physical Modelling As stated above, modelling is often performed using the physical principles that govern the system. This approach is viable whenever the physical properties of a particular subsystem are known. As even the physical laws are models and some simplifications have to be made, the physical models are not correct representations of the real system(Ljung and Glad [2004]). An advantage of physical modelling compared to system identification is that a white box model can be adapted to new similar systems by adapting the parameters accordingly. For example, a physical model of a steering system can be adapted to a new steering system with a heavier steering column by just changing the mass parameter whereas a new black box model would have to be estimated. 3.2.1 Bond Graphs Bond graphs are useful tools when performing systematic modelling. Bond graphs describe the flow of energy between components by visualizing the efforts and flow streams of the system. 3.2 15 Physical Modelling A flow of energy is represented by an arrow that indicates the direction of the energy flow. In this direction, the product of the effort, e, and flow, f , is positive. The side with the causal stroke (the right side in Figure 3.1) gives the direction for the numerical calculation for an effort variable. If all causal strokes except one are facing a serial junction, the intensity for the last one can also be calculated. The reversed relation applies to a parallel junction, if all causal strokes except one are facing away from the junction, the flow of the last one can be calculated(Ljung and Glad [2004]). e f Figure 3.1: The energy flow representation in a bond graph. The bonds of the system can be connected to elements; there are capacitive, inductive, and resistive elements as well as gyrators and transformers. A system may exchange energy with the surroundings. The surrounding energy is then referred to as a source, S. A source can be either based on the effort or flow depending on which unit is specified. A system can consist of several connection points,which are either serial or parallel. A serial point has the property that the sum of all ingoing efforts subtracted by the sum of all outgoing efforts into the connection point equals zero, which can be written ein,1 + ein,2 + · · · + ein,i − eout,1 − eout,2 − · · · − eout,j = 0. (3.3) Furthermore, all contiguous flows are equal. This can be written f1 = f2 = · · · = fn . (3.4) Analagously, a parallel connection point uses the reversed definitions, where all attached efforts are equal e1 = e2 = · · · = en (3.5) and the sum of all ingoing flows subtracted by the sum of all outgoing flows equal zero, fin,1 + fin,2 + · · · + fin,i − fout,1 − fout,2 − · · · − fout,j = 0. (3.6) Causality markings are used to specify the order of calculations. To obtain a suitable and solvable calculation, a serial connection point should have one outgoing effort and the remaining efforts should be incoming. The reversed computational order should be used for parallel connection points. For a system with consistent 16 3 General Modelling and Simulation Techniques causality markings, a state space model is easily found by choosing the flow variables of the inductive elements and effort variables of the capacitive elements as states. The steering system model is based on hydraulic and mechanical representations. In the rotational mechanical domain the effort is a torque and the flow is an angular velocity. In the translational mechanical domain the efforts are forces and the flows are velocities. The hydraulics are described with the pressure as effort and the hydraulic flow as flow. 3.3 System Identification According to Ljung [1999], three components are needed to perform system identification. The first part is analyzable data recorded with a suitable experimentation setup to capture the system behavior. The second component is a set of candidate model structures to choose from. Given a model structure, a candidate model is obtained by estimating its parameters from the data. The last component of the system identification process are rules by which candidate models are evaluated. Following the development of a model using the three components described above, the model should be validated to measure how well it explains the measured signals. 3.4 Parameter Estimation When a model structure is selected, either by physical modelling or inspecting the data, the unknown parameters, θ, need to be determined. Ljung and Glad [2004] suggests a criteria that is related to the prediction errors. At the time t − 1 the model provides a prediction ŷ(t|t − 1, θ) (3.7) of the output at time t. Using this notation the prediction error, e is e(t, θ) = y(t) − ŷ(t|t − 1, θ). (3.8) A natural intent of the modelling can be to minimize the mean square error, VN (θ) = N 1 X 2 e (t, θ). N (3.9) t=1 Using this criteria, the optimal cost is given by the parameter vector θ that best matches the measurement is obtained by minimizing VN , i.e. 3.5 17 Simulation and Solvers θ̂N = arg min VN (θ). θ (3.10) For a linear regression model ŷ(t|t − 1, θ) = θ T φ(t), where φ(t) is an arbitrary function independent of θ, minimizing the loss function, VN has an analytical solution given by the least square solution to y(t1 ) = θ T φ(t1 ) T y(t2 ) = θ φ(t1 ) . .. . y(t ) = θ T φ(t ) N N (3.11) For many non-linear model structures (ŷ(t|t − 1, θ) = h(t, θ)) where the loss function is a complex function of θ, it is much more difficult to find an analytical solution. In these circumstances the parameter estimation has to be performed numerically. A general numerical approach is described in Gustafsson [2012] as θ (i+1) = θ (i) + α (i) f (i) , (3.12) where α is a step length and f is the search direction. Basing the search direction on a first order Taylor approximation gives the Gauss-Newton algorithm defined by the recursion θ (i+1) = θ (i) + α (i) (J(θ (i) )J T (θ (i) ))−1 J(θ (i) )(ȳ − h̄(θ (i) )). (3.13) where J(x) is the Jacobian of the prediction errors, ē, with respect to θ and ȳ is a vector containing all N measurements. For a linear regression model this recursion converges in the first iteration to the least squares estimate (with α = 1). 3.5 Simulation and Solvers One important usage of models is simulation, where outputs and states of the model are determined based on initial conditions and known inputs. The method used to obtain an approximate solution depends on the type of system. In this section a brief description of solver types is given for state space and dae models based on Ljung and Glad [2004]. 3.5.1 Discretization of State Space Models To simulate a continuous state space model at f different time points t1 , ..., tf , the time-derivative of the state ẋ(t) has to be numerically approximated using 18 3 General Modelling and Simulation Techniques previous values of x(t). The simplest difference approximation is Euler’s method, in which ẋ(tn ) ≈ xn+1 − xn , h h = tn+1 − tn . (3.14) Euler’s method is generally not the most efficient one. For example, like Euler’s method, Runge-Kutta methods use more sophisticated approximations to arrive at the next state, improving accuracy at the expense of more calculations. More generally, approximations of the next state can depend on any number of previous states according to xn+1 = G(t, xn−k+1 , xn−k+2 , . . . , xn , xn+1 ), (3.15) where k is the number of states used. An important distinction is between explicit and implicit methods. In the former, the right hand side of (3.15) does not depend on xn+1 . This implies that xn+1 is found directly by evaluating the expression. In the latter, a system of equations must be solved to obtain xn+1 . Although implicit methods require more computations, they have a greater area of stability than explicit methods with the same k. Examples of multi-step methods (k > 1) are the explicit and implicit Adams methods. In general it is inefficient to keep the step length h = tn+1 − tn constant during simulations. In many cases, the dynamics shift between fast and slow. Methods determining step length automatically usually depend on the local error estimate. Stiff differential equations are characterized by widely spread time constants in the same system. To efficiently simulate stiff systems with both slow and fast dynamics, both large and small step lengths must to be managed by the solver. This is because the maximum step length is limited by stability requirements from the fast dynamics. Consequently, simulating stiff differential equations requires a trade-off between accuracy and simulation speed. Implicit methods are preferred for solvers used for stiff differential equations as they have greater stability areas. 3.5.2 Differential Algebraic Equations To simulate a general dae, one method is to approximate the derivative by using Euler’s method. This gives F( 1h (zn − zn−1 ), zn , u(tn )) = 0, . h = tn − tn−1 (3.16) Subsequently the equation can be solved for zn with the knowledge of zn−1 ; this process is then iterated for each zk , k ∈ {1, ..., N }. Obviously the accuracy of this simulation method is limited by the accuracy of 3.5 Simulation and Solvers 19 the difference approximation. To improve the difference approximation, more previous values zn−k can be used. This is called using a backward differentiation formula. Using these methods daes with index one and lower can be solved. Solving higher index daes requires other more sophisticated methods. 3.5.3 Solvers in Simulink Several solvers are implemented in the MATLAB & Simulink software. Almost arbitrary accuracy can be achieved with any solver by reducing the step size at the expense of more computations. Therefore these solvers are useful for different purposes and model types. The initial choice when choosing solver is deciding between fixed and variable step solvers. The latter type of solver is often faster than the former in accordance with the reasoning in Section 3.5.1, particularly for stiff problems. However, code generation from a MATLAB model is more complicated when using a variable step solver. The fixed and variable step solvers in Simulink are summarized in Table 3.1 and 3.2, respectively. For non-stiff problems the ode45 is generally the fastest and most accurate solver. However, for stiff problems implicit step solvers are recommended in accordance with Section 3.5.1. Primarily, ode15s is recommended for normal stiff problems. For dae problems with index < 2, a methodology similar to the backward differentiation formula described in Section 3.5.2 is implemented in both ode15s and ode23s. The numerical differentiation formulas used in ode15s are similar to but more efficient than the backward differentiation formula described in Section 3.5.2. 20 3 General Modelling and Simulation Techniques Table 3.1: Fixed step solvers in Simulink. Solver Description Order Explicit solvers ode1 ode2 ode3 ode4 ode5 ode8 Euler’s Method Heun’s Method Bogacki-Shampine Formula Fourth-Order Runge-Kutta (RK4) Formula Dormand-Prince (RK5) Formula Dormand-Prince RK8(7) Formula First Second Third Fourth Fifth Eighth Implicit solvers ode14x Newton’s method and extrapolation Variable Table 3.2: Variable step solvers in Simulink. Solver Description Order of accuracy Explicit solvers ode45 ode23 ode113 Runge-Kutta, Dormand-Prince (4,5) pair Runge-Kutta (2,3) pair of Bogacki & Shampine PECE Implementation of AdamsBashforth-Moutlon Medium Low Variable, Low to High Implicit solvers ode15s ode23s ode23t ode23tb Numerical Differentiation Formulas Fourth (NDFs) Second-order, modified Rosenbrock forFifth mula Trapezoidal rule using a "free" interEighth polant TR-BDF2 Variable, Low to Medium 4 Steering System Mechanics The mechanical connection between steering wheel and wheels is discussed in this section. First the physical laws governing the mechanics are summarized. Then a one-dimensional modelling approach is described where the subsystems are derived separately before an integrated model of the steering system mechanics is presented. 4.1 Fundamental Mechanical Principles According to Newton’s Second Law of physics, a force F on a body is given by the mass m multiplied by its acceleration a, i.e., F = ma. (4.1) The rotational equivalent of (4.1) is a torque, T , affecting a body given by the moment of inertia, J, multiplied by the angular acceleration, α, resulting in T = J α. (4.2) Since the angular acceleration is the derivative of the angular velocity, α = ω̇, (4.2) gives 21 22 4 1 ω(t) = J Steering System Mechanics Zt T (s)ds. (4.3) 0 For a torsional spring, the torsion is proportional, with constant k to the torque, Tspring . The derivative of this relationship results in ω(t) = 1 Ṫ (t). k spring (4.4) A rotational friction torque, Tf riction acting on the body is typically a nonlinear function of ω, Tf riction (t) = h(ω(t)). 4.1.1 (4.5) Moment of Inertia Model The steering system in a truck moves in three dimensions with a moment of inertia that can be written J3D Jxx = Jxy J xz Jxy Jyy Jyz Jxz Jyz . Jzz (4.6) However, in attempt to reduce the complexity of the model, a one dimensional representation of the dynamics is considered. This representation only allows each body to rotate around a given axis, commonly the z-axis, simplifying the moment of inertia to J1D = Jzz . (4.7) As the lateral dynamics being modelled concern a one-dimensional input torque to a wheel rotation in one dimension this is a reasonable simplification. 4.2 Steering Column As described in Section 2, the steering column consists of a steering wheel, an upper steering column and a lower steering column that are attached with joints. 4.2 23 Steering Column 4.2.1 Previously Developed Steering Column Models One way of modelling the upper steering column is to consider the three bodies and their corresponding moments of inertia. The torque applied to the steering wheel acts on the inertia through the steering column and is then applied to the power steering gear. The three-dimensional model developed at Scania described in Ikjema [1999] follows this approach. However, the same model constrains the relative motion of the bodies so that most torque from the driver is transferred directly to the input shaft of the power steering gear. Consequently the one-dimensional equivalent is one inertia directly connected to the torsion bar. In Tai et al. [2004] this one inertia model is used. The bodies constituting the steering column are not completely rigid but these dynamics are ignored in the model proposed by Tai et al. [2004]. Ancha et al. [2007] and Mårtensson [2005] use an approach where the steering column model is split into two bodies with separated moments of inertia, with springs and dampers inserted between the bodies. 4.2.2 Proposed Steering Column Model The moment of inertia for the steering wheel, Jsw , upper steering column, Jusc and lower steering column, Jlsc can separately be found by making the approximation that the shape of the bodies is close to that of a cylinder. The three-dimensional moment of inertia of a cylinder is J3D m(3r 2 +h2 ) 12 = 0 0 0 m(3r 2 +h2 ) 12 0 0 0 , mr 2 (4.8) 2 where m is the mass of the cylinder, r is the radius and the height is given by h. For the one-dimensional case considered here, (4.8) simplifies to J1D = mr 2 . 2 (4.9) There is friction between the steering column and its connection to the surrounding truck. This is modelled with a damper connecting one of the masses to ground. The rotational friction is typically non-linear. The full model of the three connected moments of inertia is shown in Figure 4.1. Mathematical Description The mathematical model of the steering column is developed for the full system consisting of two bodies, i.e. upper and lower steering column. As this is the most complex model of the steering column, it is easy to also find the one inertia model by eliminating the spring between the upper and lower steering column. Figure 4.2 contains a bond graph of the full steering column model. The causality of the 24 4 Steering System Mechanics Tsw Jsw + Jusc Jlsc Figure 4.1: The steering column representation. bond graph is consistent so the states can be selected as the flow variables for the moments of inertia and the effort variable for the springs. R : bsc C : ksc Se Tsw I : Jsw + Jusc s R : hµ,sc I : Jlsc s ωusc p s ωlsc Tis Tsc Se Se τl,is Figure 4.2: Bond graph of the steering column dynamics from the input torque of the steering wheel to the output torque at the output interface of the lower steering column. Consequently, the first state is the angular velocity of the steering wheel and upper steering column, ωusc . This state is calculated from the total moment of inertia of the steering wheel and the upper steering column, Jsw + Jusc , the input torque at the steering wheel, Tsw and the torque at the joint with the lower part of the steering column, Tsc , ω̇usc = 1 Tsw − Tsc . Jsw + Jusc (4.10) The second state is the torque of the spring in the joint, Tk,sc . The state equation 4.3 25 Power Steering Gear is Ṫk,sc = ksc ωusc − ωlsc , (4.11) where ksc is the torsional stiffness of the spring and ωlsc is the angular velocity of the lower steering column. The steering column torque, Tsc is equal to the torque of the spring in the joint and the friction torque given by the viscous constant bsc . Consequently (4.10) can be rewritten as ω̇usc = 1 Tsw − Tk,sc − bsc (ωusc − ωlsc ) . Jsw + Jusc (4.12) The last state in the steering column model is the angular velocity of the lower steering column, ωlsc , determined by ω̇lsc = 1 Jlsc Tk,sc + bsc (ωusc − ωlsc ) − hµ,sc (ωlsc ) − Tis − τl,is . (4.13) The state equation is dependent on the torque acting on the input shaft of the power steering gear, Tis , its associated loss torque, τl,is and a non-linear friction, hµ,sc dependent on the angular velocity of the lower steering column. The loss torque will be described in Chapter 6 and emanates from mechanical deformations and inefficiencies. 4.3 4.3.1 Power Steering Gear Previously Developed Power Steering Gear Models In the three-dimensional model developed in Ikjema [1999] the four inductive elements of the power steering gear are modelled separately; the torsion bar, worm, piston and sector shaft are modelled as four inertias or masses. However, in the one-dimensional case the torque bar and the lower steering column can be modelled as one body. Furthermore, the motions of the worm, the piston and the sector shaft are constrained which means they can be replaced with an equivalent inertia and a gear constraint. This simplification is done in Mårtensson [2005]. In Tai et al. [2004] the dynamics of the power steering gear are neglected completely; the rotational torque entering the power steering gear is directly transformed via a gear constraint to the sector shaft. 4.3.2 Proposed Power Steering Gear Model The angular displacement, between the steering column and the worm, caused by the torque bar deformation is modelled as a spring between the two bodies. This modelling alternative is used in both Ancha et al. [2007] and Mårtensson [2005]. Friction in the power steering gear is modeled with a friction force acting on the piston. In accordance with the reasoning regarding the inductive elements, 26 4 Steering System Mechanics resistive elements spread out over the piston, worm, and sector shaft can be replaced by one equivalent resistive element at the piston. There are considerable efficiency losses in the power steering gear that are modelled as asymmetries and hysteresis. Mathematical Description In this section a model including four inertias is developed. The model is then simplified to achieve causality throughout the bond graph. The efforts and flows of the mechanical forces are visualized by the bond graph in Figure 4.3. The input of the bond graph is a torque at the input shaft, transferred to the torsion bar. The flexibility of the torsion bar is modelled as a spring constant, ktb . Following Ṫtb = ktb (ωlsc − ωwo ), (4.14) the derivative of the torque that acts on the torsion bar, Ttb depends on the angular velocity of the lower steering column, ωlsc and the angular velocity of the worm, ωwo . The torsion bar torque is thus selected as a state. Se Tis I : Jtb C : ktb I : Jwo s p s ωwo Ttb I : mp `n− TF s Se Se fl,p fhyd − Fg,p I : Jss −ra TF s ωss Se Tda R : fµ,p Figure 4.3: Bond graph of the power steering gear from the input torque of the valve slide to the output torque at the sector shaft. The angular velocity of the worm is modelled according to ω̇wo = 1 (T − Ttf ,n ), Jwo tb (4.15) where Jwo is the moment of inertia of the worm and Ttf ,n is the torque of the transformer. The angular velocity of the worm is selected as a candidate state. This angular velocity is transformed in the recirculating ball mechanism to the piston, causing it to move in a translational motion with the piston velocity, vp . This is modelled as a transformer with n as the gear ratio in vp = nωwo . (4.16) 4.3 27 Power Steering Gear The corresponding translational force that causes the piston to move, Ftf ,n , can also be found through the torque of the worm using Ttf ,n = nFtf ,n . (4.17) The piston velocity is another direct candidate to use as a state since its value can be calculated from the piston mass, mp , the friction force, fµ,p , the applied hydraulics force, fhyd and the forces applied by the adjacent transformers, Ftf ,n and Ftf ,r . An extra force fl,p is subtracted from the force at the piston to account for the inefficiency of the power steering gear, described in Section 6. The power steering gear is mounted with the piston moving in an inclined plane with angle θpsg , and it is thus affected by a gravitational force, Fg,p . The piston acceleration is v̇p = 1 (F + fhyd − Fg,p − Ftf ,r − fµ,p − fl,p ). mp tf ,n (4.18) The interface between the piston and the sector shaft is modelled as a rack and pinion with radius r and acts as a transformer, Ttf ,r = rFtf ,r . (4.19) The effort is transferred from a force connected to the piston, Ftf ,r to a torque on the sector shaft, Ttf ,r . The corresponding transfer of flow, vp = rwss (4.20) allows the velocity of the piston to be transferred to the angular velocity of the sector shaft, wss . The last body of the power steering gear to be modelled is the sector shaft. Its angular velocity is calculated from its moment of inertia, Jss and the difference in torque between the incoming torque from the transformer and the torque at the droparm, Tda according to ω̇ss = 1 (T − Tda ). Jss tf ,r (4.21) Causality Conflicts and Model Adjustments It can be observed in Figure 4.3 that the bond graph of the power steering gear has causality conflicts at the piston and at the sector shaft. The serial junction at 28 4 Steering System Mechanics the piston has four causal strokes facing the junction and two causal strokes facing away from the junction. In this case, the conflict originates from the junction having two bonds trying to determine the effort. Likewise, the serial junction at the sector shaft has two bonds trying to determine the effort. These conflicts imply that a state space model cannot be derived directly from the graph. In addition to the state space equations, the static relationships from the transformers need to be included in the model. Consequently, the mathematical equations from the bond graph results in a general dae instead of a state space model. This problem can be remedied by eliminating moments of inertia and replacing them by equivalent masses. Thereby the piston, worm, and sector shaft is replaced by one equivalent mass. In order to merge the inductive elements, the moment of inertia of the sector shaft must be rescaled according to the transformation (4.19). Rewriting (4.18) as Ftf ,r = Ftf ,n + fhyd − Fg,p − fµ,p − fl,p − mp v̇p (4.22) in combination with rewriting (4.21) using (4.20) as Ttf ,r = Jss ω̇ss + Tda = Jss v̇p r + Tda (4.23) results in v̇ v̇p ( Jrss2 Jss rp + Tda + mp ) + Trda = r(Ftf ,n + fhyd − Fg,p − fµ,p − fl,p − mp v̇p ) ⇐⇒ = Ftf ,n + fhyd − Fg,p − fµ,p − fl,p . (4.24) The relation has been produced by partially removing the dae through the removal of one equation. This method is possible since the two bodies, the piston and the sector shaft are assumed to have their motions constrained one with respect to the other. Figure 4.4 shows the bond graph of the power steering gear when the piston and sector shaft are modelled as one equivalent body through the implementation of (4.24). There is still a causality conflict between the transformers. It can be solved by once again eliminating an equation and transforming the general dae into a state space model. In this case the worm is moved through the remaining transformer using (4.17) as a base and the input torque and output force of the equation need to be found. The input torque can be found with the relation that the sum of the torques at the worm node equals zero and by using (4.16), 4.4 29 Steering Linkage Se I : Jtb C : ktb I : Jwo s p s ωwo Ttb Tis I : mp + `n− TF Jss r2 s −r a TF Se Tda Se Se fl,p fhyd − Fg,p R : fµ,p Figure 4.4: Simplified model of the power steering gear from the input torque at the input shaft to the output torque at the drop arm. Ttf ,n = Ttb − Two = Ttb − Jwo ω̇wo ⇐⇒ v̇p Ttf ,n = Ttb − Jwo . n (4.25) Equation (4.24) and (4.25) can now be inserted in (4.17) to eliminate the last equation that causes the causality conflict, resulting in T v̇p 1 ⇐⇒ + mp + da + fµ,p + fl,p − fhyd + Fg,p = Ttb − Jwo r n n J J Tda T v̇p ss2 + mp + wo + = fhyd − mp gcos(θpsg ) − fµ,p − fl,p + tb . (4.26) r n r n2 v̇p Jss r2 Using (4.26) results in the causal bond graph in Figure 4.5, where the worm, piston and sector shaft are modelled as one equivalent body. The equivalent mass, meq , is described by meq = 4.4 4.4.1 Jss J + mp + wo . r2 n2 (4.27) Steering Linkage Previously Developed Steering Linkage Models In Tai et al. [2004] the entire steering system after the power steering gear is modelled as one moment of inertia with a damper affected by a non-linear friction. In Mårtensson [2005], a more complex model is developed by modelling the drop arm one inertia, followed by a transformer and then the wheels. 30 4 I: I : Jtb Se Tis Ttb p s ωlsc `n− TF Jwo n2 + mp + Steering System Mechanics Jss r2 s −r a TF Se Tda C : ktb Se Se floss fhyd − Fg,p R : fµ,p Figure 4.5: Causal representation of the power steering gear from the input torque of the valve slide to the output torque at the sector shaft. 4.4.2 Proposed Steering Linkage Model Since the sector shaft of the power steering gear is rigidly connected to the drop arm, they can be modelled as a single body with one moment of inertia. The torque that acts as an effort on these bodies is transferred to a translational force on the drag link. The proposed model is inspired by the model developed in Ikjema [1999] where the drag link is modelled as a spring and its mass is rescaled and combined with the inductive element of the drop arm. Mathematical Description A representation of the steering linkage is visualized in Figure 4.6. The input to the subsystem is the drop arm torque that originates from the power steering gear. The drop arm and drag link are modelled with an equivalent moment of inertia, 2 Jda + mdl lda , that is affected by the input torque and the force on the drag link, Fdl . Its angular velocity, ωda , is calculated according to ω̇da = 1 Tda − lda Fdl . 2 Jda + mdl lda (4.28) In this equation lda is the length of the drop arm and consequently the ratio of the torque to force transformation. Large forces act on the drag link, causing flexibility in the body [Eriksson, 1998]. The flexibility can be captured by modelling the drag link as a spring with the spring stiffness constant kdl . The force in the spring is selected as a state. The derivative of this force depends on the angular velocity of the drop arm and the angular velocity of the wheels, ωwh according to Ḟdl = kdl lda ωda − ldla ωwh . (4.29) 4.5 31 Combined State Space Model In this relation, ldla is the length of the drag link arm that transfers the translational velocity of the drag link to the rotation of the wheels. Finally, the angular velocity of the wheels is given by ω̇wh = 1 Jwh Fdl ldla − Tsa − hµ,wh , (4.30) where Jwh is the moment of inertia of the wheels including the mass of the track rod as the track rod connecting the wheels is modelled as stiff, constraining the steering angle of the left and right wheel to an equal angle. The non-linear friction of the king pin and wheels is given by hµ,wh . 2 I : Jda + mdl lda ` lda − Se Tda s ωda TF Fdl p I : Jwh R : hµ,wh s ωwh Se − ldla a TF Tsa C : kdl Figure 4.6: Causal representation of the steering linkage from the input torque of the drop arm to the output angle of the wheel. 4.5 Combined State Space Model Combining the models of the subsystems yields a complete model of the steering system mechanics. The bond graph of the complete model is shown in Figure 4.7. As the complete bond graph has integrating causality, a state space model of the system is easily derived with the flow variables of the inductive elements and the effort variables of the capacitive elements as states. For this system the states are selected as x1 ωusc x T 2 k,sc x3 ωlsc x = x4 = Ttb . x v 5 p x6 Fdl x7 ωwh (4.31) Using these states and the equations described above results a non-linear state space model, 32 4 0 ksc bsc J lsc ẋ = 0 0 0 0 1 − J +J sw usc 0 0 0 1 − Jlsc + 0 0 0 0 0 0 0 0 1 meq 0 0 1 Jlsc 0 −ksc b − J sc 0 0 0 0 0 0 0 0 0 0 0 ktb 0 0 − ntb 0 k 0 0 0 0 0 lsc n meq 0 0 l kd l da r 0 Steering System Mechanics 1 Jsw +Jusc 0 0 0 x + 0 0 0 0 kdl ldla 0 0 0 0 0 l − rmda eq 0 ldla Jwh " T # sw Tsa − J1 0 0 0 0 0 0 0 0 0 0 0 hµ,sc (x1 ) + τl,is (x3 ) fhyd (x4 , x5 ) − mp gcos(θpsg ) − fµ (x5 ) − floss (x4 ) . hµ,wh (x7 ) 0 1 wh (4.32) Jwh As can be seen in the equations, this model is a 4 dof model, where 4 moments of inertia or masses move individually. By considering the steering column rigid, as proposed above, the model from (4.32) can be simplified and two states removed. The new state vector is given by x1 ωsc x2 Ttb x = x3 = vp x4 Fdl x5 ωwh (4.33) (4.34) and the new model by 1 0 0 0 − J1 0 sc 0 Jsc ktb k 0 − n 0 0 # 0 tb 0 " Tsw lda 1 0 − 0 0 0 x + ẋ = 0 T nmeq rmeq 0 l 0 sa 0 0 kdl da 0 −k l dl dla r 0 − 1 ldla Jwh 0 0 0 0 Jwh 1 0 0 − Jsc 0 0 0 hµ,sc (x1 ) + τl,is (x1 ) 1 0 0 fhyd (x2 , x3 ) − mp gcos(θpsg ) − fµ,p (x3 ) − fl,p (x2 , x3 ) . + m eq 0 hµ,wh (x5 ) 0 0 0 0 − J1 wh (4.35) 4.6 33 Wheel Dynamics This implies removing one dof and arriving at the three dof model suggested by for example Rösth [2007] for the steering system of a truck. 4.6 Wheel Dynamics As mentioned in Section 2.3, there are important forces from wheel and tire dynamics that the steering system needs to overcome to turn the wheels. The incoming torque, Tin,wh that attempts to turn the wheel equals the sum of the torque that moves the moment of inertia, TJwh , the non-linear friction that acts between the tire and the ground, hµ,g , the non-linear between the king pin and steering knuckle, hµ,kp , and the self-aligning torque, Tsa , that acts to reduce the steering angle, i.e., Tin,wh = TJwh + hµ,g + hµ,wh + Tsa . (4.36) Hsu [2009] uses a simple but efficient way to separate the self aligning torque, separating it into a jacking torque, Tj and the total aligning torque, Ta , according to Tsa = Ta + Tj . (4.37) In this thesis, neither the total aligning torque Ta , which depends greatly on the vehicle slip angle, nor the friction between the ground and tire, hµ,g , is modelled. The jacking torque is a function of suspension geometry and is described in Hsu [2009] as Tj (δ) =Fz dhsus (δ). dδ (4.38) The vertical load on the tire is given by Fz and the change in suspension height due to steering is given by hsus . A simplified model of the derivative of the suspension height is given in Bastow et al. [2004] as eof f dhsus (δ) = sin(2θK P I )sin(δ), dδ 2 (4.39) where eof f is the lateral offset at the hub and θK P I is the kingpin inclination angle. Using (4.38) together with (4.39) gives one of the proposed models of the jacking torque that will be used later and is henceforth denoted the Sinus Model. This simple equation neglects the influence of caster and camber angle on the jacking torque. This influence is described in Wu et al. [2015] and Tagesson [2014] but not modelled in this thesis. Tsw Se s p s ωusc Se τl,is R : hµ,sc s I : Jsw + Jusc Tsc R : bsc C : ksc `n− TF Ttb p C : ktb Se I : Jlsc s vp I : meq fhyd − Fg,p − fl,p − TF r lda a R : fµ,p C : kdl Fdl p TF − ldla a s ωwh I : Jwh Tsa Se R : hµ,wh 34 4 Steering System Mechanics Figure 4.7: Bond graph of the complete steering system. 4.6 35 Wheel Dynamics An alternative way to describe the jacking torque is found in Bastow et al. [2004]. The force in the draglink induced by the jacking torque, Fdl,j is given by Fdl,j = Fz dhsus . ldla δ dδ (4.40) For small to medium angles this expression is strictly larger than 0 and it denotes the magnitude of the force. In the model, (4.40) is multiplied with sgn(δ) to find the force. Deriving the jacking torque from (4.40) by inserting (4.39) results in Tj (δ) = Fz eof f 2 sin(2θK P I ) sin(δ) sgn(δ) δ (4.41) which differs from the Sinus Model. This will be the second modelling alternative for the jacking torque used here and is denoted the Sinc Model. The difference between the Sinus Model and the Sinc Model is that the latter model applies a higher torque for lower steering angles, in comparison to a sine function that has low torques for small angles and increases the torque as the angle increases. However, the model (4.41) causes a discontinuity close to δ = 0. For computational purposes it is advisable to eliminate the discontinuity by creating a linear transition band. Notably, the two models are contradictory with very different characteristics. The reason for this disparity is not known but the models in Wu et al. [2015] and Tagesson [2014] are more similar to the Sinus model than the Sinc model. 5 Steering System Hydraulics In this section, models are developed for the hydraulic system that generates steering assistance. First the model for the hydraulics is presented and then the valve model is described. General modelling of hydraulic actuator systems for heavy vehicle applications has previously been studied in Nilsson [2014]. The modelling of the power chambers of the hydraulic system in this section is based on this thesis. Hydraulic power steering in particular has been discussed at length in Rösth [2007] and the proposed model of the valve in this thesis is inspired by Rösth [2007] and Merritt [1967]. The modelling of the hydraulics is also influenced by a previous model developed at Scania and described in Ikjema [1999]. The modelling approach used in Ikjema [1999] is applied when developing the power steering gear model in Mårtensson [2005]. 5.1 Hydraulic Model The system that provides the hydraulic assist in the power steering gear is suitably modelled as a hydraulic actuator system. In Figure 5.1 the proposed modelling principle is shown. The hydraulic force, fhyd , is generated by the pressure difference in the chambers on the two sides of the piston. Let pL and pR be the pressure in the chambers and let Ap,L = Ap,R = Ap be the area of the piston exposed to each chamber; the symmetry follows from the symmetry of the hydraulic force. Using these variables the hydraulic force can be expressed as fhyd = pL Ap,L − pR Ap,R = (pL − pR )Ap = pl Ap . 37 (5.1) 38 5 Steering System Hydraulics 𝑄2 𝑄1 𝑃𝑝 𝜓1 Pump 𝑄3 𝜓3 𝜓2 𝑄𝑝 𝑄4 𝜓4 𝑃0 𝑄𝑙 𝑃𝐿 Piston 𝑃𝑅 Figure 5.1: Respresentation of the hydraulics. When omitting the compressibility of the hydraulic fluid, the following three flows are equal: from the piston motion Ql , entering the left chamber and leaving the right chamber. In this case the chamber volume will depend on the position of the piston, xp which is bounded according to xp,min ≤ xp ≤ xp,max . (5.2) If the piston is modelled as a brick, this results in V L = V0 + A p x p (5.3) V L = V0 + A p x p (5.4) and In these equations V0 is the chamber volume when the valve angle is 0 and xp is the piston displacement from the equilibrium. Differentiating (5.3) and (5.4) results in 5.1 39 Hydraulic Model dVL = Ap ẋp = Ap vp dt (5.5) dVR = −Ap ẋp = −Ap vp . dt (5.6) and L Finally Ql = dV dt , which implies the load flow depends on the piston velocity, vp and the piston area, Ap according to Ql = vp Ap . (5.7) Each of the two power cylinder chambers are connected via the valve to the feed and return side of the hydraulic pump system. In Merritt [1967], the pressure drop over the valve and to the actuator chamber via any connection in particular can be modelled as an orifice resulting in X ∆p = ψ φ, Qf , ρQf2 2cq2 A(φ)2 (5.8) and √ Qf = ψ −1 φ, X 2cq A(φ)qX ∆p. ∆p , √ ρ (5.9) In these equation φ is the valve angle, ρ is the oil density, cq is a constant, and Av is the orifice area. This pressure drop function is henceforth denominated Ψ . The following calculations are based on Merritt [1967] and Rösth [2007]. The model for the hydraulic assist is derived by considering the hydraulic fluid incompressible and assuming constant pump flow passing through the valve; statically the pump flow and the flow passing through the valve are equal but they can vary dynamically. Furthermore the valve is modelled as symmetric, meaning that Av [φ] = A1 [φ] = A4 [φ] = A2 [−φ] = A3 [−φ]. (5.10) is fulfilled. In (5.10), the different valve areas, Av , that redirect the flow of oil are presented, these correspond to the area openings in Figure 5.1. Considering the symmetry of the hydraulic assist it is reasonable to assume that (5.10) holds. 40 5 Steering System Hydraulics This model of the system is illustrated as a bond graph in Figure 5.2. The bond graph contains a causal loop, illustrated with red lines in Figure 5.2. To avoid this issue, the bond graph can be reduced to the bond graph in Figure 5.3. The removal of the causal loop corresponds to the assumption that the pressure before the pump equals 0. This assumption seems reasonable as the pressure before the pump is small in comparison to the pressure after the pump. The modification in the bond graph does not change the value of the external variables such as the load pressure according to Ljung and Glad [2004]. From the bond graph it can be seen that Q p Q1 Q2 = Q1 + Q2 = Q3 + Q l . = Q4 − Q l (5.11) = Q p − Q l − Q2 = Q p + Q l − Q1 (5.12) This can be re-written as ( Q3 Q4 Due to the symmetry of the valve and the fact that the inflow equals the outflow, Q1 = Q4 and Q2 = Q3 . This can be further transformed into ( 2Q1 2Q2 = Qp + Ql . = Qp − Ql (5.13) Using (5.8) and (5.13), Rösth [2007] derives ρ (Qp − Ql )2 (Qp + Ql )2 − pl = p1 + p2 = 2 8cq A2v [φ] A2v [−φ] (5.14) for the pressure over the piston and ρ (Qp − Ql )2 (Qp + Ql )2 pp = p1 + p2 = 2 + 8cq A2v [φ] A2v [−φ] (5.15) for the pump pressure. As no dynamics are modelled in this physical model the result is a static relationship between the piston velocity, valve angle, and hydraulic force. These calculations will not be given in this thesis. 5.2 41 Valve Model R : ψ3 R : ψ1 p1 p Q1 s pp p Sf R : ψ2 Sf s Q2 p p2 s Q3 R : ψ4 s Ql s Q4 p p0 s Qp Figure 5.2: Bond graph of the simple hydraulic model with causal loop. 5.2 Valve Model According to Rösth [2007] there are two alternatives to valve modelling, either modelling the valve area or modelling the valve geometry. Rösth [2007] argues the importance of studying the geometry when designing a new valve but states that the area function might be more interesting when modelling an existing valve. Since this work focuses on modelling an existing valve, the approach used in this thesis is based on modelling the valve area openings. The valve is controlled by the torque in the torsion bar. When a deformation of the torsion bar occurs, this means that the angle of the input shaft is not the same as the angle of the output shaft of the power steering gear. The hydraulic force should thus increase as the deformation of the torsion bar increases in order to help steer the vehicle according to the input torque at the input shaft. The torsion at the torsion bar results in a torque equal to the spring constant multiplied by the deformation, i.e. the integral of the angular velocity according to Ṫtb = ktb ω =⇒ Ttb = ktb φ. (5.16) Ikjema [1999] presents the valve area model 3 Av [φ] = Amin + A0 e−k̃1 φ−k̃3 φ . (5.17) The area function consists of a constant and an exponential part. The design by Ikjema [1999] is motivated in part by the physical system being modelled and in part by stability concerns. As e x > 0, the valve area in (5.17) is always greater 42 5 Steering System Hydraulics R : ψ3 R : ψ1 p1 p Q1 s R : ψ2 pp p Sf s Q2 s Q3 Ql s Sf p p2 s Q4 R : ψ4 s Qp Figure 5.3: Bond graph of the simple hydraulic model with causal loop removed. than 0 when A0 and Amin are positive constants. Physically this means the valve is always slightly open. As the valve area function is always positive, the division in (5.14) does not cause any numerical issues. Numerical stability can be assured by adjusting Amin . The non-linear behavior of the hydraulics can be modelled by selecting A0 , k̃1 , and k̃3 . Since the torsion bar torque and deformation angle are proportional according to (5.16), the torque in the torsion bar can be used as a control variable instead of the angular deformation. This is convenient as the torque of the torsion bar is a state in the model shown in (3.2) and (??). Using the torque as input, the valve area function can be designed with different constants, k1 and k3 suitable to the control variable Ttb . As stated above Ikjema [1999] and Rösth [2007] model the valve as symmetrical. However Rösth [2007] states that small deviations in geometry when mounting the valve will result in the pressure curve becoming unbalanced. This asymmetry results in four different area functions, A1 [φ], A2 [φ], A3 [φ] and A4 [φ]. In production valves, such deviations are likely to occur. In (5.14), the valve area openings are modelled as symmetric. The small deviations and their resulting asymmetric pressure curves can be modelled by assigning different area functions to positive and negative torques. This solution however would only be valid for one particular power steering gear. Another method is to model a torsion bar offset, Ttb0 , to capture the unbalanced behavior in the pressure curve. The latter solution is implemented in 3 Av [Ttb ] = Amin + A0 e−k1 (Ttb −Ttb0 )−k3 (Ttb −Ttb0 ) . (5.18) 6 Data Analysis and Parameter Estimation In this chapter measurement data is used to identify parameters in the grey box physical models developed in Section 4 and 5. The data is also used to assign suitable model structures to the unknown functions. Furthermore black box models are estimated for particular subsystems. 6.1 Measurement Data The measurement data used to estimate models of different subsystems should be divided into estimation and validation data. Ideally the estimation and validation data are recorded under identical or similar circumstances. The input signal should ideally be dynamic and excite as many frequencies as possible, which means that the energy of the input should be evenly distributed across a large range of frequencies. The measurement data used to identify the system will be presented in this section. In total, six different data sets have been used in this thesis and information regarding the data sets is summarized in Table 6.1. Data Sets 1-5 are measured in a rig, under the circumstances described below. These data sets focus on capturing the behavior of the power steering gear and its hydraulics. Data Sets 1-3 are all measured statically – that is measurements are made in steady state. Data Set 6 contains measurements of a stationary vehicle. Out of these data sets, Data Set 6 is the only experiment made specifically for this thesis. Consequently, the other data sets do not necessarily have the optimal input signals or output signals for this estimation purpose. Instead the input signal to Data Sets 1-5 only consists of one single frequency; the zero frequency for Data Set 1-3 and a low non-zero frequency for Data Set 4 and 5. Furthermore, 43 44 6 Data Analysis and Parameter Estimation the components of the steering systems are not identical in all measurements, which can be a source of error. Finally, an added complication is that for Data Sets 1-5 there is not separate validation data available. Table 6.1: Measurement data. Data set Sample Time Samples No. of Series Data Set 1 0.05s 2622 1 Data Set 2 0.1s 3173 1 Data Set 3 1s 1086 1 Data Set 4 0.05s 12360 1 Data Set 5 0.1s 7479 1 Data Set 6 0.01s 5992-15212 6 6.1.1 Signals Input shaft torque and output shaft torque Input shaft torque and pump pressure Input shaft torque, pump pressure, and output shaft torque Input shaft torque and input shaft angle Output shaft torque and output shaft angle Input shaft torque, input shaft angle, and steering angle at the wheels Data Set 1 – Mechanical Efficiency The first data set consists of static measurements from the power steering gear. The input is a torque applied at the input shaft of the power steering gear and the output is a torque measured at the drop arm. To better isolate the mechanical efficiency, the hydraulic assist is deactivated. However, there is still oil in the cylinder of the power steering gear. In this scenario, the drop arm can be fixed at different angles. Figure 6.1 shows a measurement when the drop arm is fixed at zero degrees. The data set shows that the torques needed to perform a right turn and a left turn are not identical in the power steering gear mechanics. This is due to a non-linear inefficiency that needs to be modelled. 6.1.2 Data Set 2 – Valve Characteristics The second data set provides a basis for studying the valve characteristics. The input for the data set is a torque measured at the input shaft. The output is the pump pressure in the hydraulic power assist system, which in this scenario has a constant pump flow. To build up the pump pressure, the drop arm is fixed and the measurements are made under static conditions. Figure 6.2 shows the signals of the data set when the drop arm is fixed at zero degrees. 6.1 45 Measurement Data Input torque at the input shaft Normalized torque 1.5 1 0.5 0 -0.5 -1 -1.5 0 20 40 60 80 100 120 100 120 Time [s] Output torque at the droparm Normalized torque 20 10 0 -10 -20 0 20 40 60 80 Time [s] Figure 6.1: Data Set 1 with the drop arm fixed at 0 degrees and hydraulics disabled. The top plot shows the torque applied to the input shaft and the bottom plot shows the torque necessary to keep the output shaft fixed. The differences between the torques show that there are static friction and inefficiency. Input torque at the input shaft Normalized torque 1 0.5 0 -0.5 -1 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 220 240 260 280 300 Time [s] Normalized pressure Hydraulic pump pressure 1 0.5 0 0 20 40 60 80 100 120 140 160 180 200 Time [s] Figure 6.2: Data Set 2. The top plot shows the torque applied to the input shaft and the bottom plot shows the pump pressure resulting from the deformation of the torsion bar. 46 6.1.3 6 Data Analysis and Parameter Estimation Data Set 3 – Hydraulic Efficiency To further study the hydraulic power assist, a data set is available for the same scenario as Data Set 1 but with the hydraulics activated. This third data set contains static measurements of the relation between the pump pressure and the torque measured at the drop arm when the drop arm is fixed. This data set can be used to estimate inefficiencies that are not only related to the power steering gear mechanics, but the hydraulic inefficiencies as well. One plot from this data set is shown in Figure 6.3. 6.1.4 Data Set 4 – Input Analysis The fourth data set contains measurements of the input shaft torque and input shaft angle when the drop arm is disconnected from the drag link and the hydraulics are disconnected. During the experiment, the magnitude of the velocity at the input shaft is constant and low. This data set may be used to estimate one value of the viscous friction in the power steering gear and on the input shaft. The data from this data set can be seen in Figure 6.4. 6.1.5 Data Set 5 – Reversible Torque In the fifth next data set the input shaft is disconnected from the power steering gear and the drop arm is forced to move with constant magnitude of angular velocity as shown in Figure 6.5. For this data set the necessary torque and the sector arm angle are measured. Just as for the input analysis, these measurements are made with low constant velocities. 6.1.6 Data Set 6 – Stationary Vehicle Measurements In order to estimate the parameters of the steering linkage and the wheels, full system measurements are needed. However, as measurements from driving on the road are noisy and require a model of the total self-aligning torque, measurements have been made on a stationary vehicle. During the experiments, the wheels of the test vehicle are placed on low-friction turn tables that allows the wheels to turn with next to no friction from the ground. The angles of the wheels and steering column are measured, as well as the input torque applied to the input shaft. However, there are no measurements of the steering wheel torque and therefore a correct steering column model is difficult to obtain from this data set. For small angles, the left and right wheel angles are similar. However for large angles, a small deviation can be observed. A major advantage of this data set is that the self aligning torque is limited to the jacking torque and disturbances are essentially eliminated. The input signals used as input torques during these experiments are varied and separate estimation and validation data sets are available for this data set. One measurement series from this set is shown in Figure 6.6. 47 Measurement Data Normalized torque 6.1 Input torque at the input shaft 1 0.5 0 -0.5 -1 Normalized torque Normalized pressure 0 100 200 300 400 500 600 700 800 900 1000 700 800 900 1000 700 800 900 1000 Time [s] Hydraulic pump pressure 1 0.5 0 0 100 200 300 400 500 600 Time [s] Sector arm torque 1 0.5 0 -0.5 -1 0 100 200 300 400 500 600 Time [s] Figure 6.3: Data Set 3 with sector arm fixed at 0 degrees. The top plot shows the torque at the input shaft, the middle plot shows the pump pressure, and the bottom plot shows the torque necessary to keep the output shaft fixed. Input shaft torque Normalized torque 1 0.5 0 -0.5 -1 0 100 200 300 400 500 600 400 500 600 Time [s] Sector arm angle Normalized angle 1 0.5 0 -0.5 -1 0 100 200 300 Time [s] Figure 6.4: Data Set 4. The top plot shows the torque applied to the input shaft and the bottom plot shows the angle of the input shaft. 48 6 Data Analysis and Parameter Estimation Sector arm torque Normalized torque 1 0.5 0 -0.5 -1 0 100 200 300 400 500 600 700 500 600 700 Time [s] Sector arm angle Normalized angle 1 0.5 0 -0.5 -1 0 100 200 300 400 Time [s] Figure 6.5: Data Set 5. The top plot shows the torque applied to the output shaft and the bottom plot shows the resulting angle of the output shaft. Input torque at the input shaft Normalized torque 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 14 16 18 20 Time [s] Left wheel angle Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 Time [s] Figure 6.6: Data Set 6. Measurements from a stationary vehicle where the top plot shows input shaft torque and the bottom shows the steering angle of the left wheel. 6.2 49 Parameters and Functions 6.2 Parameters and Functions In this introductory section the parameters and functions are summarized. 6.2.1 Steering Column As shown in Table 6.2 almost all parameters in the model of the steering column from Section 4 are known. The notable exception is the friction and damping coefficients. Most parameters in the power steering gear are known; this is shown in Table 6.3. Just as in the steering column, no information is available on the damping and other friction forces. Even though most rig data are static, both Data Set 4 and Data Set 5 have some dynamics and can be used to estimate the friction. All parameters in the steering linkage are known and summarized in Table 6.4. As can be seen in Table 6.5, the wheel inertia and friction are unknown. Consequently these need to be estimated. Table 6.6 summarizes the parameters in the model given in Chapter 5. The density ρ in (5.14) can be found in a table. The function Av that describes the relationship between valve opening and torsion bar displacement depends on the valve design. This function will be found using system identification. The flow coefficient, cq , is unknown and has to be identified from measurement data. However, as can be seen in (5.14), cq is multiplied directly with Av and will thus not be observable as more than a scaling of the area function. Consequently the constant can be given a fixed value without affecting the solution. After this simplification, the only unknown parts of the hydraulic subsystem is the area function. This function in turn is parametrized according to (5.18) where all constants are unknowns. 6.3 Valve Characteristics As Data Set 2 shows the relationships between the torsion bar torque and pump pressure, it is used to estimate the parameters in the valve model (5.18). In Figure 6.7, the relationship between the input shaft torque and pump pressure for Data Set 2 is shown. The pump pressure increases with increasing torque magnitudes but the pump pressure is limited within an interval due to a pressure release valve, pp,min ≤ pp ≤ pp,max . (6.1) The pump pressure is given by the sum of the pressure applied on both sides of the piston as in 50 6 Data Analysis and Parameter Estimation Table 6.2: Steering column parameters. Parameter Description Estimation status Jsw Jusc Jlsc ksc Steering wheel inertia Upper steering column inertia Lower steering column inertia Steering column stiffness Calculated from geometry Calculated from geometry Calculated from geometry Known from measurements Function Description Estimation status hµ,sc Steering column friction Unknown Table 6.3: Power steering gear parameters. Parameter Description Estimation status Jwo mp n r Worm inertia Piston mass Worm to piston ratio Sector shaft radius Calculated from geometry Known Known Known Function Description Estimation status fµ,p Piston friction Unknown Table 6.4: Steering linkage parameters. Parameter Description Estimation status Jda mdl Jdla mtr kdl lda ldla Drop arm inertia Drag link mass Drag link arm inertia Track rod mass Drag link flexibility Drop arm length Drag link arm length Calculated from geometry Known Calculated from geometry Calculated from geometry Known Known Known Table 6.5: Wheels and external influence parameters. Parameter Description Estimation status Jwh eof f θK P I Fz Wheel inertia Lateral offset at hub Kingpin inclination Normal force on front axle Unknown Known Known Known Function Description Estimation status hµ,wh Wheel friction Unknown 6.3 51 Valve Characteristics ρ (Qp − Ql )2 (Qp + Ql )2 pp = pL + pR = 2 + 2 . 8cq A2v [Ttb ] Av [−Ttb ] (6.2) Figure 6.8 shows how the valve area function from (5.18) depends on the parameters A0 and Amin . These parameters can be determined from the minimal applied pump pressure in the boost curve when the piston is fixed. Logically the pump pressure will be at its lowest when the valve is in its equilibrium position, i.e. T̃tb = Ttb − Ttb0 = 0. The described scenario can be observed by simplifying (5.15) to pp,min = pp (vp = 0, Ttb 2Qp2 ρ . = 0) = 2 2 8cq (A0 + Amin ) (6.3) The parameters A0 and Amin can be determined since all the parameters except pp,min in the right hand side of r Qp ρ A0 + Amin = (6.4) 2cq pp,min are known. The minimal pressure can be approximated by the minimal measured pump pressure in the boost curve. The minimal area, Amin represents the minimal area opening of a valve. If the valve redirecting oil to the right side of the piston is completely open, the opening that redirects oil to the left hand side of the piston will be represented by Amin . As the primary purpose of Amin is guaranteeing computational stability, it can be fixed at a small value. Theoretically, Amin could be set to zero and the parameters k1 and k3 in (5.18) could be estimated accordingly. 6.3.1 Valve Offset As can be seen in Figure 6.7, there is a valve offset skewing the boost curve to the left. As noted previously, this valve offset will be different for any given power steering gear. To estimate the valve offset the measurements are sorted and low-pass filtered with a fifth order zero-phase Butterworth filter with cut-off frequency fc = 0.025H z. These filtered measurements are shown as a red line in Figure 6.9. Following the filter, a second degree polynomial is fitted to the line and shown as a black line in the same figure. Finally the input torque of the minimum of the second degree polynomial is chosen as a valid approximation of the valve offset. 6.3.2 Valve Function In Data Set 2, there is no load flow as the piston is fixed. In this case (5.14) simplifies to 52 6 Data Analysis and Parameter Estimation Table 6.6: Hydraulic parameters. Parameter Description Estimation status ρ Qp pp,min pp,max cq Ap Hydraulic fluid density Pump flow Minimum pump pressure Maximum pump pressure Flow coefficient Piston area Known Known Known Known Unknown Known Function Description Estimation status fhyd Hydraulic assist force Unknown 1.2 Normalized pump pressure 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Figure 6.7: In this figure, the relationship between the input shaft torque and pump pressure for Data Set 2 is shown. The pump pressure increases with increasing torque magnitudes. 6.3 53 Valve Characteristics Valve area opening 1 A v [-Ttb ] A v [Ttb ] A 0 + Amin Normalized valve area opening A min 0.5 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Torsion bar torque [Nm] Figure 6.8: This figure shows the valve openings area as a function of the torsion bar torque. 1.4 Measured pump pressure Filtered measurement Fitted polynomial Estimated valve offset 1.2 Normalized pump pressure 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Figure 6.9: In this figure, the relationship between the input shaft torque and pump pressure for Data Set 2 is shown. The red line is a sorted and filtered version of the measurements and the black line is a second degree polynomial fitted to the red line. 54 6 pp = pL + pR = ρ 8cq2 Data Analysis and Parameter Estimation (Qp )2 A2v (Ttb ) + (Qp )2 (6.5) A2v [−Ttb ] which in turn implies that 8cq2 pp ρQp2 = = 1 + 21 2 A2v (Ttb ) 2Av (−Ttb ) Av (Ttb )+Av (−Ttb ) . A2v (Ttb )A2v (−Ttb ) = (6.6) For any given measurement, the pump pressure is known and this means that the only unknown in (6.6) is the area function. After inserting the valve function given by (5.18), this expression can be further transformed into 0 = Ax4 + Bx3 + Cx2 + Bx + A. (6.7) where x = e−k1 (Ttb −Ttb0 )−k3 (Ttb −Ttb0 ) A = A20 − 8cq2 pp ρQp2 (A2min A20 ) = 2A2min − 8cq2 pp (2A3min A0 + 2Amin A30 ) ρQp2 8cq2 pp (A4min + A40 + 4A2min A20 ) ρQp2 B = 2Amin A0 − C 3 . (6.8) As this expression is a fourth degree polynomial it can be solved – e.g. by using the roots function in MATLAB. Obviously the equation in (6.8) will have four solutions for every measurement. To find the right solution first the complex solutions are discarded. The remaining solutions are compared with a reference solution; this reference for x is found ad hoc by tuning k1 and k3 so that they mimic the main behaviour of the valve function. The solution that is closest to the reference solution is then chosen for further manipulation and estimation. Finally, the logarithm of the chosen solution is considered a pseudo-measurement of the polynomial in the exponential of x. This measurement equation can be written y = ln(x) = −k1 (Ttb − Ttb0 ) − k3 (Ttb − Ttb0 )3 + e, (6.9) where y is the pseudo-measurements and e is a measurement error. These pseudomeasurements are shown in Figure 6.10. A first estimate for k1 and k3 is found by omitting the hysteresis and approximating Ttb by Tis . From this model k1 and k3 are found directly by linear regression. 6.4 55 Hysteresis Normalized pseudo-measurements 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 Normalized input torque Figure 6.10: On the y-axis this shows pseudo-measurements of the valve function calculated from the pump pressure in Data Set 2. The pseudomeasurements are plotted against the input shaft torque on the x-axis. This estimate is revised for some hysteresis models in Section 6.4 by developing other estimates of Ttb and using the estimate in the linear regression. 6.4 Hysteresis As noted in Section 6.1 the data from both the Mechanical Efficiency and Hydraulic Efficiency experiments are static. According to the ideal model detailed in Sections 4.5 and 5.1, this implies one input torque should always result in the same output torque, due to the lack of transients. However, as can be seen in Figure 6.11, the input-output relationship in the Mechanical Efficiency tests gives two output torques for any given input torque. This hysteresis phenomenon is also apparent in the Hydraulic Efficiency tests as is shown in Figures 6.12 and 6.13. In accordance with the definition of hysteresis phenomena, the system behaviour depends on both the current torque and its previous values. In Figure 6.14, the valve hysteresis is shown together with the time plot for the same measurement series. From this plot it can be seen that for a given input torque magnitude there are two different pump pressure curves, one when the absolute value of the input torque is increasing and a second value for decreasing torques. Upon closer analysis this conclusion is logical. When the torsion bar is being deformed, the input torque has to overwin the resistance inherit in the system – including but not limited to static friction. Analogously, when the torsion bar is returning to its natural state, the torque from the torsion bar has to overwin the 56 6 Data Analysis and Parameter Estimation 1 Normalized output torque 0.5 0 -0.5 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Figure 6.11: This figure shows the relationship between the input shaft torque(x-axis) and output shaft torque(y-axis) for Data Set 1. As the measurements are static, the two different output shaft torques for any given input shaft torque show that there is hysteresis. 1.2 Normalized pump pressure 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Figure 6.12: This figure shows the relationship between the input shaft torque(x-axis) and pump pressure (y-axis) for Data Set 2. As the measurements are static, the two different pump pressures for any given input shaft torque show that there is hysteresis. 6.4 57 Hysteresis same resistance. By performing the same analysis with respect to the other data sets, the same conclusion can be drawn. 6.4.1 Efficiency Constants It is natural to assume there are hysteresis phenomena in three places in the power steering gear: from input shaft to torsion bar, from torsion bar to piston, and from piston to output shaft. In for example Figure 6.11 it can be noted that for a given hysteresis state (e.g. increasing and positive torque) the inputoutput relationship seems approximately linear. Following this line of reasoning a reasonable model for the hysteresis is a hysteresis state dependent efficiency constant, Cef f , as shown in Tout = Cef f Tin (6.10) with Cef f = C1 C2 C3 C4 ωin ≥ 0, ωin < 0, ωin ≥ 0, ωin < 0, Tout Tout Tout Tout ≥0 ≥0 . <0 <0 (6.11) Ideally, the efficiency constants would be identical for increasing torque magnitudes, i.e. C1 = C4 , and for decreasing, C2 = C3 . However, as the measurement data demonstrates asymmetrical behaviour for positive and negative torques, all four constants are necessary. Using the efficiency constants in the three places suggested above, the loss torque and loss force in (4.32) and (4.35) can be re-written as Tl,is fl,p = (1 − Cef f ,in )Tsw = 1−Cef f ,worm Ttb n + 1 1 r 1−Cef f ,out Fdl . (6.12) With these functions and for static data – in for example Data Set 3 – (4.32) and (4.35) can be simplified to Tout = Cef f ,out r(fhyd + Cef f ,worm n1 Cef f ,in Tsw ). (6.13) Consider for example the hysteresis between the input shaft and the torsion bar and equation (6.10). Let the torque and angular velocity of the input shaft be Tin = Tis and ωin = ωis , respectively, and let Tout = Ttb be the torque in the torsion bar. If ωis and Ttb are both positive this implies the input shaft acts to increase the torque in the torsion bar. This implies that Tis has to overcome 58 6 Data Analysis and Parameter Estimation 1 Normalized output torque 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 Normalized pump pressure Input torque Figure 6.13: This figure shows the relationship between the input shaft torque(x-axis) and output shaft torque(y-axis) for Data Set 3. As the measurements are static, the two different output shaft torques for any given input shaft torque show that there is hysteresis. 1 0 -1 Pump pressure 0 50 100 150 200 250 300 200 250 300 Time (s) 1 0.5 0 0 50 100 150 Time (s) 1.2 Increasing torque Decreasing torque Pump pressure 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Input torque Figure 6.14: Data Set 2. The first plot shows the input shaft torque over time, the second plot shows the pump pressure over time, and the third plot shows the relationship between the input shaft torque(x-axis) and pump pressure (y-axis). The figure shows that a given input shaft torque gives one pump pressure when the torque magnitude is increasing and another when it is decreasing. 6.4 59 Hysteresis the resistance in the system and Cef f will therefore be smaller than 1. Similar reasoning can be made for each case. The hysteresis model in (6.10) and (6.11) has a discontinuity for ωin = 0. To prevent that high frequency oscillation of the angular velocity cause overall oscillatory behaviour, it is advisable to filter the measurements used to calculate the efficiency constants. To remove the discontinuity and improve simulation speed it is also possible to filter the efficiency constants. Combining (6.10) and (6.11) with these two filters give Tout Ċef f = Cef f Tin = Ĉef f − Cef f Kef f (6.14) with Ĉef f = C1 C2 C3 C4 ωin ≥ 0, ωin < 0, ωin ≥ 0, ωin < 0, Tout Tout Tout Tout ≥0 ≥0 <0 <0 (6.15) and ω̇in = ω̂in − ωin Kω (6.16) where Kef f and Kω are filter constants and ω̂in is the angular velocity state in the overall model. In this extended hysteresis model ωin can be considered a state describing the torque direction rather than a direct measurement of the angular velocity. Total Efficiency As Data Set 1 is both static and without hydraulic assist, (6.13) can be simplified into Tout = Cef f ,out Cef f ,worm Cef f ,in nr Tin . (6.17) Furthermore as the data is static the three hysteresis states are found by assuming that sgn(ωi ) ≈ sgn(Ṫin ) for all hysteresis states. Under this assumption the hysteresis states are found by differentiating and then filtering the input torque. The differentiation is done by the Euler approximation given in (3.14) and the filter is a low-pass, fifth order, zero-phase, Butterworth filter with cut-off frequency fc = 1H z. This same differentiation approach is used in remainder of this section. When the hysteresis states are found the values of the joint efficiency constant Cef f ,tot = Cef f ,out Cef f ,worm Cef f ,in are found by linear regression. Input Efficiency Constants and Valve Parameters Using the efficiency constants, (6.9) can be re-written as 60 6 Data Analysis and Parameter Estimation y = −k1 (Cef f ,in Tin − Ttb0 ) − k3 (Cef f ,in Tin − Ttb0 )3 + e (6.18) using the hysteresis model (6.13). The hysteresis states can be found by differentiating and smoothing Tin (as the piston is fixed sgn(ωin ) = sgn(Ṫtb ) ≈ sgn(Ṫin )). The numerical differentiation and smoothing is done as above. When the hysteresis states are found linear regression can be done on (6.18) to find k1 , k3 and the four states of Cef f ,in . Output and Worm Efficiency Constants In the Hydraulic Efficiency data set there are measurements of both input torque and output torque. For these measurements (6.13), can be written as Tout = Cef f ,tot nr Tin + Cef f ,out fhyd = = Cef f ,out Cef f ,worm Cef f ,in nr Tin + Cef f ,out fhyd . (6.19) As the total efficiency constant is known from Section (6.4.1) and the hydraulic force is known from Section 6.3, the only unknown in the equation is Cef f ,out . Analogously to the methodology used for finding the efficiency constants previously, the hysteresis constants are found by assuming that sgn(ωi ) ≈ sgn(Ṫout ) and differentiating the output torque (using the same approximation and filter as above). Finally, linear regression is used to find values for Cef f ,out conditioned on the hysteresis states. As there is now knowledge of Cef f ,out , Cef f ,in and Cef f ,tot , the final efficiency constant Cef f ,worm can now be found by using Cef f ,worm 6.4.2 = Cef f ,tot Cef f ,out Cef f ,in . (6.20) Adaptation of Hydraulic Valve Coefficients An alternative to the modelling alternative described in Section 6.4.1 is to only adapt the valve function in the hydraulics. As there is hysteresis in Data Set 1, where there are no hydraulics, this model will capture less of the system behaviour. However, as the hydraulics is already a non-linear part of the model, this modelling approach can prove advantageous from a simulation time perspective. Furthermore, it is worth noting that the hydraulic force on the piston is generally much greater than the force from the torsion bar. Considering the importance of the hydraulics, it is possible to achieve good accuracy with this model as well. One possible implementation of a valve function that compensates for the hysteresis is modelling the constants in the valve function similarly to the efficiency constants in (6.14). For both constants k1 and k3 in (5.18) this give k̇i = k̂i − ki Ki (6.21) 6.4 61 Hysteresis where k̂i = ki,1 ki,2 ki,3 ki,4 Ṫtb Ṫtb Ṫtb Ṫtb ≥ 0, < 0, ≥ 0, < 0, Ttb Ttb Ttb Ttb ≥0 ≥0 <0 <0 (6.22) and d Ṫtb = Ṫˆtb − Ṫtb Kdot . dt (6.23) Similarly to the efficiency constants, the constants ki can be found using (6.9) by linear regression. 6.4.3 Hammerstein-Wiener Models A third approach for modelling the hysteresis in the power steering gear is to estimate a Hammerstein-Wiener model. As this black box modelling approach combines a linear filter with an input and output non-linearity it might capture the system behaviour. Models with only an input non-linearity or output non-linearity belong to the subset of the Hammerstein-Wiener models known as Hammerstein models and Wiener models respectively. Although the Hammerstein-Wiener models are dynamical models, the static data from Data Sets 1-3 are used to estimate the models. This is far from optimal and can lead to erroneous conclusions. However, as only static data for the power steering gear is available, this methodology is used under the assumption that the dynamic behaviour resembles the static behaviour. To ensure the validity of the model, it is essential that the final model is validated using dynamic measurement data. The estimation of the models is performed using the System Identification Toolbox in MATLAB. Experimentation shows that using piecewise linear functions as non-linearity is a suitable model. Empirically the Wiener model Tout = hhw (Tin ) = h G(q)Tin (6.24) where h(x) = a0 x + b 0 a1 x + b 1 .. . an−1 x + bn−1 an x + bn x ≤ x0 x1 < x ≤ x2 .. . , (6.25) xn−1 < x ≤ xn xn < x G(q) is an LTI filter, and n = 4 is a good choice. A simplified explanation of why this model works is that the function works by estimating the influence of previous inputs by using the linear filter and then compensating for the asymmetrical 62 6 Data Analysis and Parameter Estimation behaviour using the non-linearity. To capture the system behaviour, two Wiener models are implemented – one on the input shaft of the power steering gear and one on the output shaft. Under the assumption that the valve constant k3 = 0 (empirically shown to be an insignificant simplification), the input shaft function, hhw,in can be estimated from the valve characteristics data set using the pseudo-measurements from (6.9). Finally, the output shaft function, hhw,out can be found using the relationship ( Ttb = hhw,in (Tin ) (6.26) Tout = hhw,out (rFout ) leading to Tout . = hhw,out (r(fhyd + n1 hhw,in (Tin ))) (6.27) for the measurements from Data Set 3 – Hydraulic Efficiency. For simulation in continuous time the estimated linear filters are converted from discrete to continuous time using the MATLAB function d2c. 6.5 Friction As the friction depends on angular velocity, it obviously cannot be identified using static data. Consequently, the Input Analysis and Reversible torque data sets are the only rig data used to estimate the friction coefficients. In addition to these data sets, the Stationary Vehicle Measurements can be used to estimate the friction. Some previous studies have been made on the friction in the steering system of Scania trucks. In Rothhämel [2013] the steering system of a truck was disassembled and then the friction was measured for varying degrees of re-assembly. Figure 6.15 shows the friction of the input shaft and the incremental friction resulting from the addition of the power steering gear and front axle respectively. The measurement of the friction is done in steady state at the steering wheel and, consequently, the friction in Figure 6.15 is not necessarily identical to the one actually acting on the piston and king pin. As can be seen in Figure 6.15, the friction on the input shaft and king pin is almost constant, i.e. Coulomb friction. On the other hand, the friction affecting the piston demonstrates a clearly viscous behaviour. 6.5.1 Linear Friction Model Considering that there is no hydraulic assist in the Input Analysis data set, the system is approximately linear. If the hysteresis is neglected, the gravitational force is considered an input, and the friction is modelled as viscous, the state space model (4.35) can be simplified into the linear representation of the system 63 Friction Normalized force 6.5 Input Friction 1 0.5 0 Normalized force 0 0.2 0.6 0.8 1 0.8 1 0.8 1 Normalized velocity Piston Friction 1 0.5 0 0 Normalized force 0.4 0.2 0.4 0.6 Normalized velocity King Pin Friction 1 0.5 0 0 0.2 0.4 0.6 Normalized velocity Figure 6.15: The top plot shows the friction friction force at different angular velocities when the steering column is isolated. The middle and bottom plot shows the incremental friction resulting from adding the power steering gear and wheels respectively. given in ẋ = Binput − J input ktb 0 1 −J 1 input 0 − kntb 1 nmeq − mp eq 0 0 1 0 Jinput 0 0 x + 0 0 0 0 0 B " T # in mp g cos(θpsg ) − 0 0 0 (6.28) meq where x1 ωsc x T x = 2 = tb . x3 vp x4 θsc (6.29) In this model the only unknowns are the gravitational force scaling – resulting from the mounting angle of the power steering gear – and the damping coefficients of the input shaft and piston. In this linear grey box model, the parameter values can be estimated numerically by e.g. the Gauss-Newton algorithm using the measurement function y= h 0 0 0 i 1 x . This algorithm is implemented in the MATLAB function greyest. (6.30) 64 6.5.2 6 Data Analysis and Parameter Estimation Non-linear Friction Models To improve the friction model, a non-linear model can be implemented. In the measurements from Data Set 6, the steering angle does not return to zero when it is released from a non-zero initial value with no steering torque. This implies there is non-negligible static friction in the wheels. In MATLAB’s Simscape the non-linear friction model described in Figure 6.16 is included. This model includes static friction with a breakaway force, Coulomb friction (constant friction), and viscous friction. The Stribeck friction is the initial decrease of the friction that can be seen in the figure. For computational reasons, the function is implemented without a discontinuity as shown in Figure 6.17 where the threshold velocity is a small number. MATLAB’s implementation of this friction model is given in fµ (FC + (Fbrk − FC ) · e−cv |v| ) · sgn(v) + Bv = −cv |v| ) v (Bvth +(FC +(Fbrk −FC )e v th |v| ≥ vth (6.31) |v| < vth where B is the viscous friction coefficient, cv is a coefficient affecting the transition velocity FC is the Coulomb friction, Fbrk is the breakaway friction, and vth is the threshold velocity. The rotational friction block is implemented analogously. Using this friction model the Coulomb and viscous friction models suggested by Qiu et al. [2001] and Wu et al. [2015] can be implemented. F Stribeck friction Breakaway friction Viscous friction Coulomb friction v Figure 6.16: General friction model used by MATLAB. The friction force (y-axis) is shown as a function of velocity (x-axis). In Ikjema [1999] the friction on the steering column of the truck is modelled as non-linear with a constant friction torque for large velocities and viscous friction 6.5 65 Friction F v Threshold velocity Figure 6.17: General friction model used by MATLAB with linear region to remove the discontinuity. The friction force (y-axis) is shown as a function of velocity (x-axis). torque for small velocities. One possible implementation of this non-linearity, is a havsin-function that depends on the angular velocity as in τµ,sc = havsin(ω, −ω̄, ω̄, −c̄, c̄) (6.32) where c̄1 , ω > ω̄1 , c̄0 +c̄1 havsin(ω, ω̄0 , ω̄1 , c̄0 , c̄1 ) = ( 2 ) + ( c̄1 −2 c̄0 ) · sin π ω̄ω−−ω̄ω̄0 − π2 , −ω̄ < ω < ω̄, . 1 0 −c̄0 , ω < ω̄0 (6.33) The maximum amplitude of the output is given by c̄. The transition velocity between linear and constant output is given by ω̄. Ikjema [1999] also develops a specific friction model for the friction on the output shaft of the power steering gear. This friction is scaled so that the friction is largest when the sector arm angle is small and decreases for larger angles. This model is given in τµ,da = A · cos(arctan(B · θda )) · havsin(ωda ). (6.34) Furthermore, Ikjema [1999] develops a model of the king pin friction. In this model the friction is modelled as Coulomb friction but, analogously with the 66 6 Data Analysis and Parameter Estimation Simscape implementation, eliminates the discontinuity by creating a linear region around zero angular velocity. The model is described in τµ,w τ̄µ,w = τ̄ v vµ,w th 6.5.3 |v| ≥ vth (6.35) . |v| < vth Proposed Friction Model Normalized force In this thesis several different friction models are tested and compared. The proposed friction models are presented in Table 6.7 and their general appearance in Figure 6.18. Firstly, viscous models are tried for all friction functions. Secondly, MATLAB’s non-linear implementation, combining Coulomb friction and viscous friction, is tried to mimic the behaviour in Figure 6.15. Thirdly, the models developed in Ikjema [1999] are tried. Input friction 2 Viscous Coulomb Coulomb-Viscous Havsin 1 0 Normalized force 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized velocity Piston friction 2 Viscous Coulomb-Viscous Havsin 1 0 0 Normalized force 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized velocity King pin friction 2 Viscous Coulomb Coulomb-Viscous 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized velocity Figure 6.18: The three plots show the proposed friction models for the input shaft, piston, and king pin respectively. The figure details how the friction force (y-axis) depends on the velocity (x-axis). The havsin-function can be used to implement both Coulomb friction and Viscous friction. This is accomplished by tuning of of ω̄0 and c̄. For Coulomb friction, the transition velocity, determined by ω̄0 should ideally by zero, but for computational reasons it should be chosen as a non-zero small number. Similarly, for semi-viscous friction the transitional velocity should be chosen large. Identification of parameters in non-linear models is done by a grid search aiming to minimize the mse cost function. The initial parameter estimates are based on the linear damping coefficients previously found. 6.6 67 External Influence Table 6.7: Proposed friction models. Model Name Description Input friction Viscous Coulomb Coulomb-Viscous Havsin Viscous friction model. Model implemented using the non-linear friction in MATLAB. Coulomb friction without discontinuity. Model implemented using the non-linear friction in MATLAB. Combines Coulomb and viscous friction. Model based on Ikjema [1999]. Viscous for low ωlsc and constant for large ωlsc . Piston friction Viscous Coulomb-Viscous Havsin Viscous friction model. Model implemented using the non-linear friction in MATLAB. Combines Coulomb and viscous friction. Model based on Ikjema [1999]. Viscous for low ωlsc and constant for large ωlsc . Friction decreases with increasing drop arm angles. King pin friction Viscous Coulomb Coulomb-Viscous 6.6 Viscous friction model. Model implemented using the non-linear friction in MATLAB similar to the model suggested by Ikjema [1999]. Coulomb friction without discontinuity. Model implemented using the non-linear friction in MATLAB. Combines Coulomb and viscous friction. External Influence In this section the influence from the self aligning torque is discussed. As the vehicle is stationary in the measurements of Data Set 6, the total aligning moment from the lateral tire forces is zero. Consequently Equation (4.36) simplifies into Tdla = J δ̈ + hµ,wh + Tj . (6.36) In this equation, only the steering angle is known from measurements. Since the steering torque in the drag link arm, Tdla , depends both on the steering system and the external forces, it cannot be directly calculated from the driver torque. However, to find initial estimates of the unknown parameters and func- 68 6 Data Analysis and Parameter Estimation tions, the steering torque can be approximated by its static relationship with the input torque when the steering angle is fixed. Furthermore, the steering angle derivatives can be found by numerical differentiation. Just as previously, the differentiation is done by the Euler approximation given in (3.14) and the filter is a low-pass, fifth order, zero-phase, Butterworth filter with cut-off frequency fc = 5H z. Using this static approximation to find the steering torque from the input torque, initial parameter estimates can be found by linear regression using either Tin = J δ̈ + beq δ̇ + Tc sgn(δ̇) + Cj sin(δ) (6.37) or Tin = J δ̈ + beq δ̇ + Tc sgn(δ̇) + Cj sin(δ) sgn(δ) δ (6.38) depending on jacking torque model. In these equations, beq and Tc is a total viscous and coulomb friction, respectively, and Cj is given by Cj = Fz eof f 2 sin(2θK P I ) (6.39) and is almost constant. Following this initial estimate, the model is continuously refined – by a grid search with a mse cost function – together with the friction models where beq and Tc are used as guidance for the friction models. As there is no measurement data where any of these constants can be separated they are all estimated at the same time. 7 Results In this chapter the models previously developed are validated and compared using measurement data. As the measurement data presented in Chapter 6 are either for the power steering gear in isolation or for the total steering system, this chapter will be structured according to these two categories. In accordance with the purpose of this thesis, models are compared for accuracy and computability. These concepts are measured using the normalized root mean square error, nrmse, nrmse = kŷ − yk ky − ȳk (7.1) as a proxy for accuracy and simulation time as a proxy for computability. The computability is only calculated for the complete vehicle simulations as the simulation time for static measurements does not correspond well to the times for dynamic simulations. As it is the relative simulation time and relative nrmse that is interesting for each model combination, the nrmse and simulation time will be normalized for each comparison table. In practice this implies that the fastest simulation will be given relative simulation time 1, the most accurate model will be given relative nrmse 1, and the others are scaled accordingly. Although normalizing the nrmse and simulation time facilitates the comparison of model combinations, it also makes it impossible to determine whether the model is good in absolute terms. In order to show the absolute accuracy and efficiency, the absolute nrmse and simulation time is given for the final model and when nrmse is discussed in the text. 69 70 7.1 7 Results Static Behaviour and Hysteresis In this section the result from the estimation and validation using Data Set 1-3 is presented. Unfortunately, there is no validation data for these data series and comparisons are therefore made using estimation data. 7.1.1 Static Gain As stated previously, the gear ratio of the power steering gear nr , is approximately constant and known for small to medium angles. Ideally this gear ratio, as a static gain, would describe the input-output relationship for the static data of Data Set 1. However, as stated in Section 6.4 this model has poor predictive ability due to torque losses in the power steering gear. Similarly, the least squares estimate of the static gain has low predictive ability due to the hysteresis phenomena. The measurements are compared to simulations using these two choices of static gains in Figure 7.1. The simulations using ideal and estimated gain have nrmse of 0.464 and 0.19 respectively. Simulation of Data Set 3, for the same ideal case with no torque loss, results in Figure 7.2. As can be seen in the figure, there are considerable torque losses in the system and the nrmse for this data is 0.492 for the pressure and 0.448 for the output torque. 7.1.2 Efficiency Constants Following the estimation approach suggested in Section 6.4.1, efficiency constants are estimated for (6.14), (6.15), and (6.16) and subsequently used for simulation. Simulation using the identified efficiency constants, without taking any system dynamics into account, results in the result showed in Figure 7.3. As the figure shows, there is generally good correspondence (nrmse 0.049) between the measurement data and simulation. The notable exception is given by the maximum value for the torque where the efficiency constant changes value due to a change of state. Considering that the efficiency constants transition will be smoother when the steering dynamics change gradually, this is natural. To assess the models true fit to measurement data, the system is simulated dynamically where the input is kept constant between samples. In this simulation the steering dynamics results in a natural low pass filter and the simulation showed in Figure 7.4 with a slightly improved nrmse of 0.043. In accordance with the method described in Section 6.4.1, efficiency constants are estimated for the pseudo measurements of the torsion bar torque. As can be seen in Figure 7.5, the efficiency constants capture the hysteresis behaviour in the pseudo measurements with nrmse 0.06. Furthermore, the simulated pump pressure closely resembles the measurements but with considerable noise for large pump pressures resulting in a nrmse of 0.268. Following identification of all three sets of efficiency constants, the model is simulated for Data Set 3. The result of this simulation is shown in Figure 7.6. As 7.1 71 Static Behaviour and Hysteresis 1.5 Measurement True gear ratio Estimated gear ratio 1 Normalized torque 0.5 0 -0.5 -1 -1.5 0 20 40 60 80 100 120 Time [s] Normalized pressure Figure 7.1: Simulations of Data Set 1 with static gains. Simulation with the true gear ratio and an estimated gear ratio is compared to the estimation data. The y-axis shows the output torque, normalized with the maximum absolute value of the measurement data, as a function of time. 1 0.5 0 -0.5 Measurement Simulation -1 0 200 400 600 800 1000 1200 800 1000 1200 Time [s] Normalized output torque 1.5 1 0.5 0 -0.5 -1 -1.5 0 200 400 600 Time [s] Figure 7.2: Simulation of Data Set 3 with the ideal static model. The simulated output pressure and output torque is compared to the estimation data in the top and bottom plot, respectively. The y-axis is normalized with the maximum absolute value of the measurement data. 72 7 Results Normalized output torque 1 0.5 0 -0.5 -1 Measurement Simulation -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Normalized output torque 1 0.5 0 -0.5 -1 -1.5 0 20 40 60 80 100 120 Time [s] Figure 7.3: Simulations of Data Set 1 with efficiency constants but no system dynamics. The simulated output torque (y-axis) is compared to the estimation data with input torque on the top x-axis and time on the bottom x-axis. The y-axis and top x-axis are normalized with the maximum absolute value of the measurement data. Normalized output torque 1 0.5 0 -0.5 -1 Measurement Simulation -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Normalized input torque Normalized output torque 1 0.5 0 -0.5 -1 -1.5 0 20 40 60 80 100 120 Time [s] Figure 7.4: Simulations of Data Set 1 with efficiency constants and with system dynamics. The simulated output torque (y-axis) is compared to the estimation data, with input torque on the top x-axis and time on the bottom x-axis. The y-axis and top x-axis is normalized with the maximum absolute value of the measurement data. 7.1 Static Behaviour and Hysteresis 73 the figure shows, the simulation has reasonable fit to the measurement data with pressure nrmse 0.263 and output torque nrmse 0.204. However, considering that there is no separate validation data for these data sets, it is possible that the model suffers from over-adaptation to this measurement set or this type of input and spectral content. This risk is partially mitigated by the fact that this model has a physical interpretation; the efficiency constants correspond to the resistance in the steering system. 7.1.3 Hydraulic Adaptation As an alternative to modelling the non-linear inefficiencies at the input shaft and at the sector shaft of the power steering gear, the valve can be modelled to compensate for the inefficiencies. This approach leads to different constants estimated in (6.21), (6.22), and (6.23). Simulations with the hydraulic adaptation results in Figure 7.7 with good results, 0.249 nrmse to measurement data of the hydraulic pressure and 0.154 nrmse of the output torque. However due to variations in the valve, there is a risk of over-adaptation to a specific power steering gear or to the data set itself. 7.1.4 Wiener Model Analogously to the efficiency constant approach, first the hysteresis of the input is estimated using the pseudo-measurements in (6.24) and (6.25). Simulating the estimated model gives Figure 7.8. The simulation has good correspondence with the estimation data but there is a slight offset leading to too low pump pressure for positive torques and too high for negative torques. This gives a nrmse compared to the pseudo measurements of 0.113 and nrmse 0.293 compared to the pressure data. Using the input function from Figure 7.8 and the data from Data Set 3, the output hysteresis is estimated and then simulated. The simulation results can be found in Figure 7.9. Obviously the offset from the input function is still apparent in this simulation with nrmse 0.293. However, it is interesting that the simulation has very good nrmse 0.048 compared to the output torque. This implies the output function compensates for the bias introduced by the input function. Due to the lack of validation data, it is more probable that the Wiener approach suffers from over-adaptation to the measurement data than, for example, the efficiency constants; the functions derived from the Wiener model lack the clear physical interpretation of the efficiency constants. In order to diminish this risk, each linear filter is selected with fewer than 4 poles and 4 zeros. It is also problematic that the dynamic Wiener models are estimated using static data. As there is no other data available, this method is still used which increases the risk for over-adaptation. 74 7 Results Pseudo-measurements 3 Measurement Simulation 2 1 0 -1 -2 -3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Normalized input torque Normalized pump pressure 1.2 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 Time [s] Normalized pressure Figure 7.5: Simulation of Data Set 2 with efficiency constants. The simulated pump pressure (y-axis) is compared to the estimation data in the bottom plot and the simulated pseudo-measurements (y-axis) are compared to the estimation data in the top plot with input torque on the x-axis. The y-axis and top x-axis are normalized with the maximum absolute value of the measurement data. 1 0.5 0 Measurement Simulation -0.5 0 200 400 600 800 1000 1200 800 1000 1200 Normalized output torque Time [s] 1 0.5 0 -0.5 -1 0 200 400 600 Time [s] Figure 7.6: Simulation of Data Set 3 with efficiency constants. The simulated output pressure and output torque is compared to the estimation data in the top and bottom plot, respectively. The y-axis is normalized with the maximum absolute value of the measurement data. 7.1 75 Static Behaviour and Hysteresis Normalized pressure 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 Time [s] Normalized output torque 1 0.5 0 -0.5 Measurement Simulation -1 0 200 400 600 800 1000 1200 Time [s] Figure 7.7: Simulation of Data Set 3 with the Hydraulic Adaptation model. The simulated output pressure and output torque is compared to the estimation data in the top and bottom plot, respectively. The y-axis is normalized with the maximum absolute value of the measurement data. Pseudo-measurements 3 Measurement Simulation 2 1 0 -1 -2 -3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Normalized input torque Normalized pump pressure 1.2 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 Time [s] Figure 7.8: Simulation of Data Set 2 with the Wiener Model. The simulated pump pressure (y-axis) is compared to the estimation data in the bottom plot and the simulated pseudo-measurements (y-axis) are compared to the estimation data in the top plot with input torque on the x-axis. The y-axis and top x-axis are normalized with the maximum absolute value of the measurement data. 76 7.1.5 7 Results Comparison of Hysteresis Models for Static Data Table 7.1 summarizes the result from the simulations of Data Set 3. As can be seen from the simulation results, there is truly non-negligible torque losses in the power steering gear that need to be modelled. Best fit to the pressure data is obtained through the hydraulic adaptation, followed by the efficiency constant and the Wiener approach. Regarding the output torque, the best fit is obtained from the Wiener approach even though both the efficiency constants and the hydraulic adaptation give acceptable results. For all these approaches there is a risk for over-adaptation to static data in general and these measurement series in particular. As the Wiener models is a black box approach and estimated using static data, this is extra vulnerable to overadaptation. Table 7.1: Comparison of Hysteresis Models for static data from Data Set 3. The nrmse is normalized. 7.2 Modelling Approach Pressure nrmse Torque nrmse Ideal Model Efficiency Constants Hydraulic Adaptation Wiener 1.98 1.06 1.00 1.18 9.31 4.29 3.20 1.00 Dynamic Behaviour and Full Vehicle Steering System Model For the full vehicle data from Data Set 6, both simulation time and fit is calculated. However, as there is only measurement of input torque and steering angle, is is not possible to isolate the friction, self-aligning torque, or hysteresis model. To avoid an unnecessary amount of figures, all possible modelling alternatives are not shown. Generally all modelling options but one are kept constant for any given comparison. For example, the hysteresis models are compared using one model for the self-aligning torque – not every self-aligning torque model. However, where considerable interdependence between modelling options is noticed, the best combination for each alternative is shown. An example is the hysteresis models and input friction models. Both these modelling options aim to model the resistance on the input shaft and from the grid search it can be seen that the hysteresis models work best with different input friction models. Since Data Set 6 is designed specifically for this thesis, the input signals are varied and excite a large range of frequencies. To ensure that the models developed are not good only for one specific input signal, all comparisons are made using the two distinct measurement series Data Set 6.1 and Data Set 6.2 (belonging to Data Set 6). As these measurement series, shown in Figure 7.10 and 7.11 respectively, 7.2 77 Dynamic Behaviour and Full Vehicle Steering System Model contain fundamentally different frequencies, this decreases the risk that models are over-adapted. To further decrease this risk, the measurement series are split into estimation and validation data. 7.2.1 Comparison of Hysteresis Models for Dynamic Data In Table 7.2, the results of the simulations with different hysteresis models are summarized and the results are plotted in Figure 7.12 and 7.13. As stated above there is considerable interdependence between the hysteresis model and input friction model and therefore the input friction has been adapted to each hysteresis model individually. The best fit is obtained from the ideal model but the efficiency constants and the Hydraulic Adaptation model also has reasonable fit. The Wiener model in no way manages to follow the dynamic data for neither data set and it must be concluded that the model is both over-adapted to static data in general and the specific estimation data in particular. Regarding the simulation time, the ideal model, which contain the fewest states, naturally simulates the fastest. However, both the efficiency constant model and the Hydraulic Adaptation model have acceptable simulation speeds as well. In addition to having less than satisfying fit to the measurement data, the Wiener model is the slowest due to a larger number of states. Table 7.2: Comparison of Hysteresis Models for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach Ideal Model Efficiency Constants Hydraulic Adaptation Wiener 7.2.2 Data Set 6.1 Relative Time nrmse 1.00 1.97 1.40 5.24 1.00 1.00 1.10 3.30 Data Set 6.2 Relative Time nrmse 1.04 1.00 1.32 2.71 1.00 1.01 1.36 1.87 Comparison of Models for the Jacking Torque In Section 4.6, two models for the jacking torque were presented, (4.38) and (4.41). As there is no total aligning moment for these measurements this is the complete models. In Table 7.3 the results are presented. As can be seen in the table and Figure 7.14, the Sinc Model, which applies a stronger torque at low steering angles, achieves slightly better fit for one of the data sets and the Sinus Model for the other. The simulation time for the two models are the same. However, to achieve this fit with the Sinus Model, the constants have to be selected larger than they should be physically. Likewise, the parameter in the Sinc Model needs to be adapted to the measurement data but to a lesser extent. 78 7 Results Normalized pressure 1.2 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 Time [s] Normalized torque 1 0.5 0 -0.5 Measurement Simulation -1 0 200 400 600 800 1000 1200 Time [s] Figure 7.9: Simulation of Data Set 3 with the Wiener model.The simulated output pressure and output torque is compared to the estimation data in the top and bottom plot, respectively. The y-axis is normalized with the maximum absolute value of the measurement data. Input torque at the input shaft Normalized torque 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 14 16 18 20 Time [s] Left wheel angle Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 Time [s] Figure 7.10: Data Set 6.1. The measurement is made for dynamic data with activated hydraulics on a stationary vehicle. The input torque is applied at the input shaft and is displayed in the top plot. The resulting wheel angle of the left wheel is displayed in the bottom plot. The y-axis is normalized with the maximum absolute value of the respective signal. 7.2 79 Dynamic Behaviour and Full Vehicle Steering System Model Input torque at the input shaft Normalized torque 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 14 16 18 20 Time [s] Left wheel angle Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 Time [s] Figure 7.11: Data Set 6.2. The measurement is made for dynamic data with activated hydraulics on a stationary vehicle. The input torque is applied at the input shaft and is displayed in the top plot. The resulting wheel angle of the left wheel is displayed in the bottom plot. The y-axis is normalized with the maximum absolute value of the respective signal. Data Set 6.1 Normalized angle 1 0.5 0 -0.5 -1 -1.5 0 2 4 6 8 10 12 14 16 18 20 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0 -1 Measurement Ideal model Efficiency constants Weiner -2 -3 0 2 4 6 8 10 12 Time [s] Figure 7.12: Comparison of Ideal, Efficiency Constant, and Wiener models for dynamic data. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. 80 7 Data Set 6.1 1 Normalized angle Results Measurement Ideal model Hydraulic adaptation 0.5 0 -0.5 -1 -1.5 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0.5 0 -0.5 -1 -1.5 0 2 4 6 8 10 Time [s] Figure 7.13: Comparison of the Hydraulic Adaptation model and Ideal model for dynamic data. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. Data Set 6.1 Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0.5 0 -0.5 -1 Measurement Sinus Sinc -1.5 -2 0 2 4 6 8 10 12 14 Time [s] Figure 7.14: Comparison of jacking torque models for dynamic data. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. 7.2 81 Dynamic Behaviour and Full Vehicle Steering System Model Table 7.3: Comparison of Jacking Torque Models for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach Sinus Model Sinc Model 7.2.3 Data Set 6.1 Relative Time nrmse 1.00 1.04 1.22 1.00 Data Set 6.2 Relative Time nrmse 1.00 1.17 1.00 1.14 Comparison of Friction Models Input Friction Following iterative use of the grey-box estimation approach for Data Set 4 in (6.28), described in Section 6.5.1, the estimation converges and results in the simulation output in Figure 7.15. As is apparent in the figure the identified coefficients give good nrmse 0.094 to estimation data. However, as the friction depends on the angular velocity which is constant for this measurement set, this result is only valid for this particular angular velocity. It is worth mentioning that the friction estimate from this approach also compensates for the hysteresis that is omitted from this linear model. When the estimated Viscous model is used to simulate Data Set 6 that has faster dynamics, the estimated friction is too high. Similarly, iterative estimation of viscous models for Data Set 6.1 and Data Set 6.2 shows that Data Set 6.1 with faster dynamics requires less damping than Data Set 6.2. Consequently, the friction over angular velocity ratio has to decrease as the angular velocity increases. This behaviour can be simulated both using the Havsin model, (6.32) suggested by Ikjema [1999] and a Coulomb-Viscous friction according to (6.31). Iterative estimation of Havsin Input Friction Models and Coulomb-Viscous Models result in the model output shown in Figure 7.16 and summarized in Table 7.4. The Coulomb-Viscous model demonstrates the same, but less expressed, tradeoff between the fast and slow dynamics as the Viscous model. To a certain extent this is also true for the Havsin Model but by fine-tuning the parameters much of this behaviour can be eliminated. The Coulomb Model, as indicated by Rothhämel [2013], causes the opposite trade off; more friction is required for the less dynamic Data Set 6.2. The parameter values in the Coulomb-Viscous and Havsin models depend heavily on the hysteresis model. In comparison to the ideal model, the efficiency constant model requires less Coulomb friction. This is natural as the torque losses modelled by the efficiency constants are probably caused, to a large extent, by friction. Piston Friction Table 7.5 presents a comparison of friction models that are estimated for dynamic data. As can be seen in Table 7.5 and as Rothhämel [2013] suggested, a com- 82 7 1.5 Results Measurement Simulation Normalized angle 1 0.5 0 -0.5 -1 0 100 200 300 400 500 600 700 Time [s] Figure 7.15: Simulation of Data Set 4 with a Linear Friction model. The simulated input shaft angle is compared to estimation data. The y-axis is normalized with the maximum absolute value of the measurement data. Data Set 6.1 Normalized angle 1 Measurement Viscous Coulomb Coulomb-Viscous Havsin 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 Time [s] Figure 7.16: Simulation of Data Set 6 with different input friction models. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. 7.2 83 Dynamic Behaviour and Full Vehicle Steering System Model Table 7.4: Comparison of Input Friction Models for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach Viscous Model Coulomb Model Coulomb-Viscous Model Havsin Model Data Set 6.1 Relative Time nrmse 1.00 1.17 1.16 1.01 1.85 1.49 1.70 1.00 Data Set 6.2 Relative Time nrmse 1.00 1.40 1.88 1.82 1.26 1.00 1.46 1.23 pletely viscous model gives good fit to validation data. Adding Coulomb friction implemented according to the Matlab function, does not improve the accuracy of the simulation and increases simulation time. Similarly, adapting the Havsin model for sector arm friction developed in Ikjema [1999] to the piston does not improve the simulation either. The results from these experiments are shown in Figure 7.17. Table 7.5: Comparison of piston friction models for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach Viscous Model Coloumb Model Coloumb-Viscous Model Havsin Model Data Set 6.1 Relative Time nrmse 1.00 1.59 1.12 1.17 1.00 1.08 1.00 1.46 Data Set 6.2 Relative Time nrmse 1.00 1.59 1.10 1.03 1.10 1.00 1.10 1.22 King Pin Friction When adapting the viscous wheel friction model to the estimation data it is clear that the wheel friction has a rather small impact on the model accuracy. Consequently, exchanging the friction model for a Coulomb friction model does not greatly affect the simulation – except for increasing the simulation time. The results for the different friction models can be found in Table 7.6 and Figure 7.18. Table 7.6: Comparison of king pin friction models for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach Viscous Model Coulomb Model Data Set 6.1 Relative Time nrmse 1.00 1.39 1.02 1.00 Data Set 6.2 Relative Time nrmse 1.00 1.57 1.00 1.10 84 7 Data Set 6.1 1.5 Normalized angle Results Measurement Viscous Coulomb-Viscous Havsin 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 Time [s] Figure 7.17: Simulation and comparison to Data Set 6 with different piston friction models. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. Data Set 6.1 Normalized angle 1.5 Measurement Viscous Coulomb 1 0.5 0 -0.5 -1 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 1 0.5 0 -0.5 -1 0 2 4 6 8 10 Time [s] Figure 7.18: Simulation and comparison to Data Set 6 with different king pin friction models. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. 7.3 85 Conclusions 7.2.4 Comparison of Solvers In order to investigate how the choice of solver affects the accuracy and computability, the model was simulated using three different solvers, ode5, ode45, and ode15s. As can be seen in Table 7.7 these tests unanimously suggest ode15s is the most suitable solver. The simulation using ode45 achieved the same simulation result but required much longer time. When using ode5, the simulation either reached maximum simulation time without finishing or failed, depending on step size. This result implies that the state space model is stiff and requires an implicit variable solver. Table 7.7: Comparison of Solvers for dynamic data from Data Set 6. The nrmse is normalized. Modelling Approach ode5 ode45 ode15s 7.3 Data Set 6.1 Relative Time nrmse NaN 62.8 1.00 NaN 1.00 1.00 Data Set 6.2 Relative Time nrmse NaN 127 1.00 NaN 1.00 1.00 Conclusions Throughout this chapter there have been both accurate and less accurate models. In this section the necessary choices and trade-offs are made that maximize the accuracy without requiring too long simulation time. In Section 4.5 two steering column models are developed with one and two moments of inertia respectively. As the torque for all measurement sets is measured below the split between the upper and lower steering column, no comparison of accuracy can be made using the measurement data available. In order to make such a comparison it would be necessary to measure the torque applied at the steering wheel in addition to the torque applied at the input shaft. As the added flexibility is believed to have negligible impact on the model accuracy and some impact on the computability, the simpler one moment of inertia steering column model is selected. This decision is also supported by the review of mechanical models in Dell’Amico [2013]. Regarding the hysteresis modelling, the best choice for dynamic data is considered the ideal model. Although at a clear disadvantage compared to other models for static data, the ideal model has the best fit and simulation time for dynamic data. The hysteresis that the ideal model fails to capture for static simulations is compensated using the friction model for dynamic data. An additional benefit of the ideal model is that it has fewer parameters and is less likely to be overadapted than the hysteresis models. Although the jacking torque models have fundamental differences they both man- 86 7 Results age to capture significant model behaviour. However, even though the Sinus model has better theoretical support, its parameters need to be one order of magnitude larger than they should be physically to achieve this. On the other hand, the Sinc Model has less support in the literature but its parameters are in the right order of magnitude and it performs slightly better experimentally. Consequently, as neither of the alternatives completely constitutes a physical representation of the system, the Sinc model is chosen for its superior performance empirically. Considering that the input friction model presented in Ikjema [1999] has better fit than the alternatives and shorter simulation time, it is clearly the best choice to use the Havsin friction model for the input friction. However, for the piston and the wheels, the viscous model is relatively accurate and significantly faster than the alternatives. Consequently, this is the modelling alternative of choice. Finally, the choice of solver is facilitated greatly by the superiority of ode15s. The final simulation result is shown in Figure 7.19. This model is rather accurate for both validation series with nrmse 0.307 and 0.499, respectively. Furthermore, the model can be simulated with acceptable speed as 100 seconds of data require a simulation time of approximately 2 seconds. As the model to a large extent is developed by physical modelling – as opposed to system identification – the model can be adapted for some limited variations of the steering system and truck configuration without having to estimate a completely new model. On the other hand, other modelled parts are specific to this steering system. An example is the valve offset which will be distinct for every single power steering gear. It is also possible that the estimated efficiency constants are only valid for this particular power steering gear. 7.3 87 Conclusions Data Set 6.1 Normalized angle 1 0.5 0 -0.5 Measurement Simulation -1 0 2 4 6 8 10 12 14 16 18 20 12 14 16 18 20 Time [s] Data Set 6.2 Normalized angle 0.4 0.2 0 -0.2 -0.4 -0.6 0 2 4 6 8 10 Time [s] Figure 7.19: Simulation of the final model and comparison to Data Set 6. The simulated left wheel angle (y-axis) is compared to validation data for two different data sets in the two plots. The y-axis is normalized with the maximum absolute value of the measurement data. 8 Summary In this chapter the results from Chapter 7 are discussed and compared with the purpose of the thesis. Additionally, future work is proposed and discussed. The rest of this chapter is structured along these two topics. 8.1 Discussion The underlying objective of this thesis is deepening the understanding of steering systems in general and for heavy duty vehicles in particular. More specifically this objective leads to the purpose of this thesis which is developing an accurate and efficient physical model of the steering system. In pursuit of this purpose, the modelling effort has included the basic mechanical connection and various non-linear phenomena such as hysteresis, friction, and hydraulic assist. As a result of the modelling, there is a simulation model achieving the necessary accuracy. By also identifying the most important model constituents and simplifying the model, efficiency in the form of simulation time is achieved. However, although a relatively accurate and efficient model of the steering system has been identified, this is not enough to use for development of e.g. lka. While driving, the steering system is merely a subsystem of the total truck which in turn can be considered a subsystem of the truck-and-environment system. Considering this, the model describes the steering system but requires interaction from other models or sensors to serve its primary purpose. Regarding the model itself, it meets the requirements of accuracy and computability. However, the model is only better than the rest of the identified models – it can in no way be considered optimal. It is possible that better results could be 89 90 8 Summary obtained from the same modelling approach with better conditions, including but not limited to a consistent measurement set-up, more dynamic data from isolated subsystems, and more validation data. There are also other modelling approaches that could be attempted. This would result in other models with their own advantages and drawbacks. Finally, it is worth discussing the generalizability of the results obtained. The data used for estimation is from a few selected steering systems and the extent to which these results and methodologies can be transferred to other trucks of the same and different types is uncertain. It is probable that the model developed in this thesis is best adapted to the particular truck studied. However, it is also probable that the model would work rather well for a similar truck and that the same methodology would be directly applicable. For different steering systems, it is likely that some adaptations are necessary to find a well-working simulation model, but it is also likely this thesis would greatly facilitate that modelling effort. In summary, this thesis has resulted in a well-functioning simulation model and thus fulfilled its purpose. In doing so it has also granted insight into the mechanics and function of steering systems in general. Consequently, this thesis has made its own contribution to the emergence of autonomous driving. 8.2 8.2.1 Future Work Suggestions for Model Accuracy Improvement Some of the work done in this thesis has been developed based on measurement data from isolated power steering gears where the power steering gears are not identical in all measurements. This is a source of error that could be eliminated by using rig data from the same power steering gear as the one from which full vehicle data is recorded. The developed model would be further improved by studying different power steering gears and their corresponding full vehicle data. The development process would also be simplified and model accuracy would be increased by studying more subsystems separately before identifying the fully assembled system. In the developed model, the power steering gear has been identified separately. Using the same methodology, the steering column and steering linkage could also be isolated and identified. The drag link can advantageously be assigned sensors that measure its position. As a result the force in the drag link can be derived. This would also mean that the steering system dynamics could be measured when decoupling all external effects and applying incremental modelling. The used rig data does not contain a wide frequency spectrum on the input signal since the rig data sets are not primarily designed for this purpose. Measuring separate estimation and validation data is also suggested to design models that are not over-adapted to the specific data set. 8.2 Future Work 8.2.2 91 Model Expansion Throughout the development of the model, little emphasis has been put on finding a good vehicle model that models the self aligning torque, weight distribution during turns and steering angle to yaw rate conversion. A complete model of the lateral dynamics needs to consider such external influences. The developed model in this thesis is a good model for a stationary vehicle without friction acting on the tires. How the wheel-tire friction affects a stationary vehicle as well as vehicles at higher velocities need to be studied. The authors recommend that additional measurements are recorded and used for such estimations. These measurements should advantageously include the input torque, the wheel angle, the drag link position and yaw rate for a vehicle driving on a flat surface at varying velocities. Finally, the model can be further generalized and expanded by studying the steering system of different vehicle configurations. For example, the king pin friction likely depends on the weight on the front axle – the importance of which has not been studied in this thesis. 92 8 Summary Bibliography Santosh Ancha, Abhijit Baviskar, John R. Wagner, and Darren M. Dawson. 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